The complement of A ∪ B is the set of all elements not in A ∪ B.
If x is some element of A ∪ B, then x is either in A or B (or both).
So if x is some element of the complement of A ∪ B, that means x does not belong to A or B. In other words, x is not in A and x is not in B.
So (A ∪ B)' = A' ∩ B', where the ' symbol denotes set complement.
For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6} are both subsets of U = {0, 1, 2, 3, 4, 5, 6, 7}, then
A ∪ B = {1, 2, 3, 4, 5, 6}
→ (A ∪ B)' = {0, 7}
Put another way,
A' = {0, 5, 6, 7}
B' = {0, 1, 2, 7}
→ A' ∩ B' = {0, 7}
and we see that both (A ∪ B)' and A' ∩ B' are the same set.