We need to be defined as so that many mathematical formulae work. For example we would like to work when ie Also we require that the formula for the number of ways of choosing objects from is valid for ie is valid when

Things need to work when we extend our definition of the factorial via the gamma function.

The above gives and so we require since

One definition of the factorial that is more general than the usual

$$ N! = N\cdot(N-1) \dots 1 $$

is via the gamma function, where

$$ \Gamma(N) = (N-1)! = \int_0^{\infty} x^{N-1}e^{-x} dx $$

This definition is not limited to positive integers, and in fact can be taken as the definition of the factorial for non-integers. With this definition, you can quite clearly see that

$$ 0! = \Gamma(1) = \int_0^{\infty} e^{-x} dx = 1 $$

If you are starting from the "usual" definition of the factorial, in my opinion it is best to take the statement $0! = 1$ as a part of the definition of the factorial function, as anything else would require proofs using the factorial to include special cases for $0!$ and $1!$. It's a definition that is consistent and makes our lives easier.