Let me start off by saying that I am not a mathematician, and that I will be using some pseudo-mathematical terms in the interest of writing something more akin to simplified English rather than accurate mathematical jargon.
The question as stated is:
Why does any non-zero number to the zeroth power equal one?
To answer this question lets first talk about what is meant by "zeroth" power.
"zeroth power" refers to exponentiation. To understand why the zeroth power works the way it does it's important that we properly define exponentiation.
Exponentiation is the act of raising a number to the power of another number. That's actually not too helpful because now you need to know what "raising to the power" means.
...
But first, lets talk about multiplication.
Multiplication is the act of adding a number ($a$) some other number ($b$) of times ($a \times b$).
$$2 + 2 + 2 = 2 \times 3$$
This is all well and good, but when we talk about mulitplying by $0$ we need to know what number to put on the left hand side:
$$? = 2 \times 0$$
The base for addition is $0$. It's the additive identity. Every addition equation may be implicitly started with $0$. This means that above, two times three is actually:
$$0 + 2 + 2 + 2 = 2 \times 3$$
In this form certain behaviors become quite clear:
$$0 + 2 + 2 + 2 = 2 \times 3$$
$$0 + 2 + 2 = 2 \times 2$$
$$0 + 2 = 2 \times 1$$
$$0 = 2 \times 0$$
Negatives also make sense, because instead of adding numbers, you do the opposite, you un-add (often called "subtraction"):
$$0 = 2 \times 0$$
$$0 - 2 = 2 \times -1$$
$$0 - 2 - 2 = 2 \times -2$$
...Ok, with all that in mind, now it's time to look at exponentiation.
Exponentiation is the act of multiplying a number ($a$) some other number ($b$) of times ($a ^ b$)
$$2 \times 2 \times 2 = 2 ^ 3$$
This is all well and good but when we talk about "raising to the power of 0" we need to know what number to put on the left hand side:
$$? = 2 ^ 0$$
The base for multiplication is $1$. It's the multiplicative identity. Every multiplication equation may be implicitly started with $1$. This means that above, two to the power of three is actually:
$$1 \times 2 \times 2 \times 2 = 2 ^ 3$$
In this form certain behaviors become quite clear:
$$1 \times 2 \times 2 \times 2 = 2 ^ 3$$
$$1 \times 2 \times 2 = 2 ^ 2$$
$$1 \times 2 = 2 ^ 1$$
$$1 = 2 ^ 0$$
Likewise, negatives also make sense, because instead of multiplying numbers, you do the opposite, you un-multiply (often called "division"):
$$1 = 2 ^ 0$$
$$1 \div 2 = 2^{-1}$$
$$1 \div 2 \div 2 = 2^{-2}$$
Note that these patterns hold regardless of the base:
$$n \times n \times n \times 1 = n^3$$
$$n \times n \times 1 = n^2$$
$$n \times 1 = n^1$$
$$1 = n^0$$
$$1 \div n = n^{-1}$$
$$1 \div n \div n = n^{-2}$$
$$1 \div n \div n \div n = n^{-3}$$