Why does any nonzero number to the zeroth power = 1?

Here's a good heuristic that helped me feel better about it back in the day:

Consider any number $x$ to any power. We will choose, say, $5^3$.

$5^3 = 125$. Divide both sides by $5$. You get $5^2=25$, which we know. Again. Then you get $5^1=5$. We usually leave exponents of $1$ off but we'll keep it here. Do you see what happens? The power goes down by $1$ each time. We can do it again. Then following our pattern, $5^0=1$. But $5$ wasn't special!

You can do this with any real number. Of course, for irrationals it can get a little fuzzy.

By the way, we can keep going. Divide by $5$ again. You get $5^{-1}=\frac{1}{5}$. I hope this helps!

Clearly the value of $x^0$ is a matter of definition. A simple way to think about this that only involves non-negative integer powers is to notice that for positive integers $a$ and $b$ you want $$ x^{a+b}=x^a \, x^b. $$ If you were to define $x^0$ you would likely want to have that hold for $b=0$, that is $$ x^{a+0}=x^a \, x^0 \rightarrow x^{a}=x^a \, x^0 \rightarrow x^0=1. $$ Now there are other ways of getting to this that involves the definition of the exponential function (as in infinite polynomial series) but the above is the simplest.

If you would think of it backwards, you would see that, in the case of 2, for example,

$2^4$ =16

$2^3$ = 8

$2^2$ = 4

$2^1$ = 2

So, it is logical to continue dividing by two,and reach the conclusion that $2^0$ = 1. This is true for negative powers, and is so because it is the most logical way to define powers outside of the natural numbers.