Static Range Sum Queries - Why Prefix Sums

Build a prefix sum array where prefix[i] = sum of elements from index 1 to i. This takes $O(n)$ time once. Now any range sum [a,b] = prefix[b] - prefix[a-1].

Why? prefix[b] includes everything up to b. Subtracting prefix[a-1] removes everything before a. You've converted q queries of $O(n)$ each into q queries of $O(1)$ each. Total: $O(n + q)$ instead of $O(n \times q)$.