The right-hand side is the number of binary strings of length 2n, consisting of 0s and 1s.

For k < n, the term of index k on the left is the number of such strings in which the digit in position n + k + 1 occurs there for the (n + 1)st time. When k = n, it is the number of strings in which there are n 0s and n 1s.

This proves the equality.