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.