So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1. (src = scienceline.org)
0^1 is 0 because anything to the power of 1 is itself (since the 1 means how many times, 1 time is just the number given). Thus 0^1 is the base number, in which is 0.
Anything being multipled by 0 such as your example 0 raised to the first power is 0 this is because exponents are repeated multiplication. Since anything raised to the first power is just equal to its base since you aren't multiplying anything.