I'm currently trying to understand the process of blowing-up, and a few things strike me as a little difficult to get an intuitive understanding of what's happening.

The current problem is on self intersection.

Why does the exceptional divisor I get by blowing up at a point $\mathbb{P}^1(\mathbb{C})\times\mathbb{P}^1(\mathbb{C})$ have self intersection -1? A mathematically abstract answer would be nice, but an included intuitive explanation of negative self-intersection would really take the cake.