The statement is wrong: the probability should be , i.e. every residue of modulo is equally likely.

There is a strong symmetry between the variables, and this is the key.

Consider the simplest case to better understand it: if are random with equal probability to equal 0 or 1, and independent of each other, then is 0 if has same parity as and 1 else. However, by the case with n-1 variables, the parity of is 0 or 1 with same probability. Using independence, we conclude that the parities match with probability 1/2.

The general case is the same. You can procede by induction: you have (equalities are modulo N) (using independence) and the conclusion is immediate since each probability is by assumption or induction hypothesis.