AWK, 48 bytes

{printf"%"$1+NF-1"s\n",/^[0-9]/?substr($0,2):$0}

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Cubically, 69 66 bytes

RBR3B~(+50<7?6{+54>7?6{-002+7~?6{(@5*1-1/1)6}}}(-6>7?6&@7+70-4~)6)

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Explanation:

First we do this initialization:

RBR3B

To set up this cube:

   533
   004
   000
411223455441
311222331440
311222331440
   555
   555
   200

The most important thing about this cube is that face 5 sums to 32, which is the value required to print spaces. Coincidentally, it also happens to be fairly short for all other computation. After that is done:

~( . . . )                                    Takes the first input, then loops indefinitely

  +50<7?6{+54>7?6{-002+7~?6{(@5*1-1/1)6}}}    Handle leading digit:
  +50<7?6{                               }    If input is greater than 47 ('0' is 48)
          +54>7?6{                      }     And input is less than 58 ('9' is 57)
                                              Then input is a digit
                  -002+7                      Set notepad equal to value of input digit
                        ~                     Take next input (only convenient place for it)
                         ?6{           }      If the notepad isn't 0
                            (        )6       While the notepad isn't 0:
                             @5                 Print a space
                               *1-1/1           Decrement the notepad by one
                                              Leading digit handled

     (-6>7?6&@7+70-4~)6                       Handle rest of line:
     (               )6                       While the notepad isn't 0:
      -6>7?6&                                   Exit if End of Input
             @7                                 Print the next character
               +70-4                            Set notepad to 0 if it was a newline
                    ~                           Take the next character

><>, 60 bytes

!^i:0(?;::"/")$":"(*0$.
v"0"-
>:?!v1-" "o
;>:o>a=&10&?.i:0(?

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How It Works:

..i:0(?;... Gets input and ends if it is EOF
...
...
...

.^......::"/")$":"(*0$. If the inputted character is a digit go to the second line
...                     Else go to the fourth
...
...

....        If it was a digit
v"0"-       Subtract the character "0" from it to turn it into the corresponding integer
>:?!v1-" "o And print that many spaces before rejoining the fourth line
...

.^..               On the fourth line,
....               Copy and print the input (skip this if it was a digit)
....v              If the input is a newline, go back to the first line.
;>:o>a=&10&?.i:0(? Else get the input, ending on EOF

Japt (v2.0a0), 11 10 bytes

Japt beating Jelly and 05AB1E? That doesn't seem right!

r/^\d/m_°ç

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Explanation

Implicit input of string U

r/^\d/m

Use Regex replace (r) all occurrences of a digit at the beginning of a line (m is the multiline flag - the g flag is enabled by default in Japt).

_

Pass each match through a function, where Z is the current element.

°

The postfix increment operator (++). This converts Z to an integer without increasing it for the following operation.

ç

Repeat a space character Z times.

Implicitly output the resulting string.

PHP, 83 chars

preg_replace_callback('/^\d/m',function($m){return str_repeat(' ',$m[0]);},$argv);

Cubically, 82 bytes

R3D1R1D1+0(?6{?7@7~:1+2<7?6{+35>7?6{:7-120?6{(B3@5B1-0)6}:0}}}?6!@7~-60=7&6+4-3=7)

Note: This will not work on TIO. To test this, use the Lua interpreter with the experimental flag set to true (to enable conditionals). There's currently a bug with conditional blocks on the TIO interpreter. When using the TIO interpreter, you should replace ?6! with !6 and &6 with ?6&, which keeps the byte count the same.

R3D1R1D1          Set the cube so that face 0 has value 1 and the rest of the values are easy to calculate

+0                Set the notepad to 1 so that it enters the conditional below
(                 Do
  ?6{               If the notepad is 1 (last character was \n or start of input)
    ?7@7              Output the current character if it's \n
    ~                 Get the next character
    :1+2<7?6{         If the input is >= '0'
      +35>7?6{          If the input is <= '9'
        :7-120            Set the notepad to the input - '0'
        ?6{               If the notepad isn't 0
          (                 Do
            B3@5              Output a space
            B1-0              Subtract 1 from notepad
          )6                While notepad > 0
        }                 End if
        :0              Set notepad to 1
      }                 End if
    }                 End if
  }                 End if

  ?6!@7             If the notepad is 0 (did not attempt to print spaces), print current character

  ~                 Get next character
  -60=7&6           If there is no more input, exit the program
  +4-3=7            Check if current character is \n, setting notepad to result
)                 Repeat forever

This isn't as short as the other Cubically answer, but I thought I'd give this a try anyway :D

R, 138 128 bytes

-9 bytes thanks to CriminallyVulgar

n=readLines();for(d in grep("^[0-9]",n))n[d]=gsub('^.?',paste0(rep(' ',eval(substr(n[d],1,1))),collapse=''),n[d]);cat(n,sep='
')

This is pretty bad, but it's a bit better now... R is, once again, terrible at strings.

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Java 8, 105 99 97 93 bytes

Saved few more bytes thanks to Nevay's suggestion,

s->{int i=s.charAt(0);if(i>47&i<58)s=s.substring(1);while(i-->48)s=" "+s;System.out.print(s);}

05AB1E, 10 bytes

v0y¬dićú},

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Husk, 15 13 bytes

-2 bytes thanks to @Zgarb

mΓo+?oR' i;±¶

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Uses the same technique as @Jonathan Allan

Explanation

             ¶  -- split input into a list of lines
m               -- apply the following function to each line
 Γ              --   deconstruct the string into a head and a tail
  o+            --   prepend to the tail of the string ...
    ?      ±    --     if the head is a digit (n)
     oR' i      --       the string of n spaces
                --     else
          ;     --       the head of the string
                -- implicitly print list of strings line-by-line

Ruby, 24 21+1 = 25 22 bytes

Uses the -p flag. -3 bytes from GB.

sub(/^\d/){"%#$&s"%p}

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Gema, 21 characters

\N<D1>=@repeat{$1;\ }

Sample run:

bash-4.4$ gema '\N<D1>=@repeat{$1;\ }' <<< 'foo bar foo bar
> 1foo bar foo bar foo bar
> 2foo bar foo bar foo bar foo bar
> 
> --------v
> 8|
> 8|
> 80
> 8,
> 7&'
foo bar foo bar
 foo bar foo bar foo bar
  foo bar foo bar foo bar foo bar

--------v
        |
        |
        0
        ,
       &

JavaScript (ES8), 38 37 bytes

a=>a.replace(/^\d/gm,a=>''.padEnd(a))

I don't think it can be improved much more.
Saved 1 byte thanks to Shaggy - Use ES8 features.

V, 9 bytes

ç^ä/x@"é 

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Explanation

ç  /      ' On lines matching
 ^ä       ' (Start)(digit)
    x     ' Delete the first character
     @"   ' (Copy Register) number of times
       é  ' Insert a space

Python 2, 98 74 67 65 bytes

-24 bytes thanks to Jonathan Allan. -7 bytes thanks to Mr. Xcoder.

for i in open('f'):print' '*int(i[0])+i[1:]if'/'<i[:1]<':'else i,

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Takes input in the file named f.

Python 2, 76 72 68 bytes

-4 bytes thanks to @ovs!

@DeadPossum suggested switching to Python 2, which saved 4 bytes too.

Just thought it's nice to have a competitive full program in Python 2 that does not explicitly check if the first character is a digit. This reads the input from a file, f.

for i in open('f'):
 try:r=int(i[0])*" "+i[1:]
 except:r=i
 print r,

Try it online! (courtesy of @ovs)

Haskell, 63 bytes

unlines.map g.lines
g(x:r)|x<';',x>'/'=(' '<$['1'..x])++r
g s=s

Try it online! The first line is an anonymous function which splits a given string into lines, applies the function g to each line and joins the resulting lines with newlines. In g it is checked whether the first character x of a line is a digit. If this is the case, then ['1'..x] yields a string with length equal to the value of the digit x and ' '<$ converts the string into as many spaces. Finally the rest of the line r is appended. If x is not a digit we are in the second equation g s=s and return the line unmodified.

Pyth,  16  15 bytes

jm.x+*;shdtdd.z

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Explanation

jm.x+*;shdtdd.z   - Full program that works by reading everything from STDIN.

             .z  - Read all STDIN and split it by linefeeds.
 m               - Map with a variable d.
  .x             - Try:
     *;shd           - To convert the first character to an Integer and multiply it by a space.
    +     td         - And add everything except for the first character
            d        - If the above fails, just add the whole String.
j                 - Join by newlines.

Let's take an example that should be easier to process. Say our input is:

foo bar foo bar
1foo bar foo bar foo bar
2foo bar foo bar foo bar foo bar

The program above will do the following:

• .z - Reads it all and splits it by newlines, so we get ['foo bar foo bar', '1foo bar foo bar foo bar', '2foo bar foo bar foo bar foo bar'].

• We get the first character of each: ['f', '1', '2'].

• If it is convertible to an integer, we repeat a space that integer times and add the rest of the String. Else, we just place the whole String. Hence, we have ['foo bar foo bar', ' foo bar foo bar foo bar', ' foo bar foo bar foo bar foo bar'].

• Finally, we join by newlines, so our result is:

  foo bar foo bar
   foo bar foo bar foo bar
    foo bar foo bar foo bar foo bar
  

Jelly, 19 bytes

Ḣ⁶ẋ;µ¹µḣ1ẇØDµ?
ỴÇ€Y

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-5 bytes total thanks to Jonathan Allan's comments and by looking at his post

Explanation

Ḣ⁶ẋ;µ¹µḣ1ẇØDµ?  Main link
             ?  Ternary if
                if:
       ḣ1       the first 1 element(s) (`Head` would modify the list which is not wanted)
         ẇ      is a sublist of (essentially "is an element of")
          ØD    "0123456789"
                then:
  ẋ             repeat
 ⁶              ' '
Ḣ               n times where n is the first character of the line (head)
   ;            concatenate the "beheaded" string (wording choice credited to Jonathan Allan)
                else:
     ¹          Identity (do nothing)
    µ µ     µ   Link separators
ỴÇ€Y            Executed Link
Ỵ               Split by newlines
  €             For each element,
 Ç              call the last link on it
   Y            Join by newlines

Perl 5, 13 + 1 (-p) = 14 bytes

s/^\d/$"x$&/e

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Jelly, 19 bytes

V⁶ẋ
Ḣǹe?ØD;
ỴÇ€Yḟ0

A monadic link taking and returning lists of characters, or a full program printing the result.

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How?

V⁶ẋ - Link 1, make spaces: character (a digit)
V   - evaluate as Jelly code (get the number the character represents)
 ⁶  - a space character
  ẋ - repeat

Ḣǹe?ØD; - Link 2, process a line: list of characters
Ḣ        - head (get the first character and modify the line)
         -   Note: yields zero for empty lines
     ØD  - digit characters = "0123456789"
    ?    - if:
   e     - ...condition: exists in? (is the head a digit?)
 Ç       - ...then: call the last link as a monad (with the head as an argument)
  ¹      - ...else: identity (do nothing; yields the head)
       ; - concatenate with the beheaded line

ỴÇ€Yḟ0 - Main link: list of characters
Ỵ      - split at newlines
 Ç€    - call the last link (1) as a monad for €ach
   Y   - join with newlines
    ḟ0 - filter out any zeros (the results of empty lines)

Retina, 9 bytes

m`^\d
$* 

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Python 3, 95 bytes

lambda y:'\n'.join(re.sub('^\d',lambda x:' '*int(x.group()),z)for z in y.split('\n'))
import re

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-4 bytes by stealing the regex idea from ThePirateBay