We can prove this via straightforward counting techniques and some elementary algebra. You want to first count all the A sets and then for each of those A sets you want to count all possible B sets such that their intersection comes out to be exactly equal to 1. So first we choose A set from n elements. Then we choose the element we want to be common in the both sets. Then from the remaining elements we just choose the B set. So from this we get the following summation.
We can now observe that the formula for the binomial theorem can be used to simplify the summation with