Jelly, 12 bytes

L!_UÆ¡$œ?J’U

Try it online!

Explanation

We can basically see the two lists (the input and the output) as encoding an integer; the input encodes an integer in factorial base, and the output encodes an integer as a permutation. Luckily, Jelly has builtins that are already very close to both of these encodings, so it's simply a matter of writing small pieces of code to convert to an integer, then back to the other encoding.

L!_UÆ¡$œ?J’U
   U           Reverse {the input}
    Æ¡         and convert from base factorial to integer;
  _   $        subtract that from
L!             the factorial of the length of {the input};
       œ?      then take the nth permutation of
         J     [1,2,...,l], where l is the length of {the input},
          ’    subtract 1 from every elevent,
           U   and reverse it

In the case of base factorial, we can observe that the first element of the list must be 0, the second can be 0 or 1, the third must be 0/1/2, and so on. Thus, we have to reverse the input in order to get its elements into the normal writing order for base conversion.

Additionally, for the relative orders of the factorial conversion and the permutation conversion to match up with the operation that insertion sort uses, we need to make two adjustments: reversing the sequence of the permutations, and reversing the order of the output list. Reversing the output list is easy enough, needing only a U at the end of the program. To reverse the sequence of permutations, we subtract from the factorial of the input length (this works because the base factorial produces a number in the range 0 to (length!-1), whereas the permutations are numbered by Jelly from 1 to length!, producing an implicit off-by-one that cancels out the off-by-one that you normally get when subtracting a permutation index from a factorial).

Mathematica, 92 bytes

Permute[Range[l=Length@#]-1,(c=Cycles@{#}&)@{}©##&@@c[i-0~Range~#[[i]]]~Table~{i,l,1,-1}]&

Pure function taking a list of nonnegative integers as input and returning a list of nonnegative integers. The above code contains a ©, which is incorrect: it's a placeholder for the 3-byte symbol U+F3DE, which Mathematica represents by a circle with a dot in it, and which represents composition of permutations.

c=Cycles@{#}& defines a function that converts a list of integers into a Cycles object representing a permutation; for example, c[{3,4}] is the transposition swapping elements 3 and 4 of a list. c[i-0~Range~#[[i]]]~Table~{i,l,1,-1}] takes the input list and generates the permutations necessary to undo the insertion sort. Then c@{}©##&@@ composes all of these permutations together, starting with the identity permutation c@{}. Finally, Permute[Range[l=Length@#]-1,...] applies this permutation to the 0-indexed list of appropriate length.

Python 2, 79 bytes

a=input();j=len(a);b=range(j)
for i in b:j-=1;b.insert(j,b.pop(j-a[j]))
print b

Works by generating the moves in reverse

JavaScript (ES6), 73 bytes

m=>(a=m.map((_,i)=>j=i)).map(_=>a.splice(j,0,a.splice(j-m[j--],1)[0]))&&a

Test cases

let f =

m=>(a=m.map((_,i)=>j=i)).map(_=>a.splice(j,0,a.splice(j-m[j--],1)[0]))&&a

console.log(JSON.stringify(f([0,0,0])));       // -> [0,1,2]
console.log(JSON.stringify(f([0,1,0,1])));     // -> [1,0,3,2]
console.log(JSON.stringify(f([0,0,0,0,0,5]))); // -> [1,2,3,4,5,0]
console.log(JSON.stringify(f([0,1,2,3])));     // -> [3,2,1,0]
console.log(JSON.stringify(f([0,1,1,1])));     // -> [3,0,1,2]
console.log(JSON.stringify(f([0,1,1,2,1,0]))); // -> [4,0,2,1,3,5]

JavaScript (ES6), 69 65 bytes

a=>a.reverse(b=[...a.keys()]).map(o=>b.splice(~o,1)[0]).reverse()

Annoyingly both input and output are in the wrong order. Edit: Saved 4 bytes thanks to @Arnauld.