Substring Sum Set

Jelly, 6 bytes

ṫJḣ"ḅ1

Try it online! or verify all test cases.

How it works

ṫJḣ"ḅ1  Main link. Argument: A (array)

 J      Index; yield the 1-based indices of A.
ṫ       Tail; map k to the postfix of A that begins with the k-th element.
  ḣ"    Vectorized head; for each k in A, truncate the corr. postfix to length k.
    ḅ1  Convert the resulting slices from base 1 to integer.

Python, 40 bytes

f=lambda x:x and[sum(x[:x[0]])]+f(x[1:])

Test it on Ideone.

Excel, 21 bytes

=SUM(OFFSET(A1,,,A1))

Open a new spreadsheet, put the test values in column A. Enter the formula in B1 and double click the cell handle to ride the range.