Think of yourself at the center of a circle with a series of concentric rings. If you are vaccinated with a 90% effective vaccine, your chance of catching the bug is 10% from anyone in the first ring. However, if everyone in the first ring is vaccinated, the chance of catching the bug from anyone in the second ring is only 1% (i.e 0.1*0.1). Thus with enough people vaccinated even a crappy vaccine works pretty well.

Perhaps a rather simplistic model, but somewhat illustrative.