One easy way to see this: if d is a factor of n, then n/d is a factor of n (since d divides n evenly). So, for any integer, we have pairs of factors (d and n/d) and thus an even number of factors, except in the case when d and n/d are the same.

When are they the same? When d = n/d, or d^2 = n (n is a perfect square). Therefore, only perfect squares have an odd number of factors (2 * the number of pairs of factors + 1).