What happens (as described by larry and wnjj pretty well, among others) is a consequence of Kirchoff's Current Law, which is a foundational priciple in electrical circuit analysis. It states that at every point in time the sum of currents flowing into a node in a circuit is zero--or, stated another way, that the current into the node and out of the node is exactly equal. (Current flowing into the node has an opposite sign from that flowing out of the node.)
Taking the neutral to be the 0V reference, when one leg is in its positive cycle the other is in its negative cycle, so the current due to one leg is going into the node at the neutral and the current due to the other leg is going out of that node. The difference of these two currents, which is just the amount they are unbalanced with respect to each other, is what flows through the neutral connection.
It might make more sense if you think of a plumbing analogy. If you have a tee in a pipe system, with some amount of water flowing into one side of the tee and the same amount flowing out the other side, the net flow through the third opening must be zero.