Why is two to the power of zero equal to binary one?
Because we want $2^{m+n} = 2^m \cdot 2^n$, and if $n = 0$ this requires that $2^0 = 1$. More combinatorially, $a^b$ is the number of functions from a set with $b$ elements to a set with $a$ elements, and there is exactly one function from the empty set to any other set (the empty function). This is the same reason that $0! = 1$.
Any number (except zero) to the zero power equals one. There are many ways to justify it. Easiest (to me) is to make the laws of exponents work. We know that $a^b/a^c=a^{b-c}$. Now put b=c