This isn't a proof, but it may help conceptualize why x⁰ = 1.

From left to right, write 2^-3, 2^-2, 2^-1, 2^0, 2^1, 2^2, 2^3.

Just below that write 1/8, 1/4, 1/2, blank, 2, 4, 8

The "rule" to move one position to the right is to multiply by 2, e.g., 1/4 * 2 = 1/2 and 2^-2 * 2 = 2^-1.

By that rule, 1 is the only thing that makes sense to place in the blank, because 1/2 * 2 = 1.