Answer by Edwin Chen:
Here's a geometric (but non-rigorous) way to think about it:
x^n is the volume of an n-cube with side length x, aka the number of unit n-cubes you need to make an n-cube with side length x. For example, eight unit cubes make a cube of side length two (2^3 = 8).
In zero dimensions, everything is a point. In particular, the unit 0-cube and the 0-cube of side length x are both just points. Since it only takes one point to make another point, we have x^0 = 1.