Why is the gambler's fallacy a fallacy?

I have always been perplexed by a seeming paradox in probability that I'm sure has some simple, well-known explanation We say that a "fair coin" has "no memory." At each toss, the odds are once again reset, at 50:50. Hence, the " gambler's fallacy." After 10 heads, the odds of another head are still 50:50. And the same, after 20, 40, 80 heads. Yet, we know that the series will converge, pretty quickly, upon an equilibrium of heads:tails. How can both be true? Is there something in the physical series of tosses that "remembers"? Is there necessarily some slightly better chance of a tails, after 10 heads? How can logic reconcile the absolute randomness of particular events, with a general law of convergence? It raises the larger issue of what sort of "causality" probability is. Why don't fair coin tosses add up?