The intuitive explanation is that the effect of a single sample on the average diminishes as you take more samples. So, hand-waving a bit, let's assume it's true that over a large number of trials you would expect the average to converge to 0. You just tossed a coin and got heads, so you're at +1. The average of (1 + 0*n)/(n+1) still goes to 0 as n grows bigger and bigger.
That skips over the distinction between "average" and "probability distribution", but those are nuances are probably better left for a proof of the central limit theorem.
That skips over the distinction between "average" and "probability distribution", but those are nuances are probably better left for a proof of the central limit theorem.