The best intuitive explanation I have come across for why dividing by N the sum of squared deviations from the sample mean underestimates the population variance is that the sample mean "follows" the sample; i.e. the sample almost always deviates from its own mean less than it deviates from the population mean (and it never deviates more). This is the source of the bias frodo72 alluded to.
This intuitive argument only shows that simply taking the sample average of squared deviations from the sample mean will underestimate the population variance, but it does not at all prove that N/(N − 1) is the right correction factor. I don't know of an intuitive argument for this, but a nice rigorous derivation can be found
here.
