Why X0 equals zero?

He is not assuming X₀ to be 0, he is defining X₀ to be 0.

The formula for the amplitude of the coefficients is undefined at coefficient 0. If you try to calculate it from the formula given, you will arrive at zero, infinity, or arguably any number in between.

It's rather like the behaviour of the Fourier transform of a pulse response. You can fit the ratio sin(kx)/kx to the entire function, except at x=0, where you again have a 0/0. The important thing here is that the thing you are modelling is defined at x=0, and the arithmetical formula apparently doesn't fit, so is usually defined for x=0 when you first meet this formulation.

There are techniques to determine the value of f(x)/g(x) where both f and g tend to zero at the same value of x on to a more thorough footing, by expanding f and g in polynomials in x, and cancelling terms. However, that detail is beyond the scope of the question you've been set. You might however try doing it for sin(x)/x as a fun exercise for yourself.