From my knowledge, it has to do with the multiplicative identity. In short, any number to the zero power is the product of no numbers whatsoever, which is just 1.

A more intuitive thought process can be explored using division and the subtraction rule of exponents. Any number divided by itself is equal to 1. For example, 2 divided by 2 is equal to 1. Now, what if we divide 2¹ by 2^1? We know that 2¹ is is 2, so 2¹ divided by itself is also 1 (2 divided by 2 equals 1).

Continuing, if we want to subtract 2¹ by 2^1, we get 2⁰ (using the subtraction rule of exponents) which is 1. Combining the division and subtraction rule of exponents deems that the only possible answer is 1, not non-existence. If something is non-existent in mathematics, it is proved by contradiction using an existence theorem.

Not sure if this makes complete sense, but the proof is the zero power rule is usually investigated in a math class called real analysis. This can be a rigorous proof, so most teachers/mathematicians just say “believe me, its 1.” I could go into more of that proof, but it won’t serve well unless you know some rigorous math about integers and their classification.