It doesn't happen because the same mutation happens again. It happens because people (and other species) pass the mutation on to their children. My father, like every father, had ~75 new mutations when he was born -- that is, 75 places where his DNA was different than the DNA of either of his parents. He passed his DNA on to my brother and me, so he passed on those mutations -- those different bits of DNA.
But passing on DNA is a crapshoot. Like everyone, my father had two copies of each chromosome, so for each of the 75 places where his DNA had changed, he had a 50% chance of passing on the mutated version and a 50% chance of passing on the unmutated version. For some of those 75 mutations, my brother and I each got the mutated copy, for others only one of us did, and for others neither did. So when my father was born, each of the 75 had one copy in the population. Today, some of them have two copies, some one and some zero. In short, the frequency of the mutations changed from one generation to the next, purely because of chance.
The same process will go on every generation. In the next generation, some of the 75 have 3 copies, some 2, some 1 and some 0. And it's going on in everyone else in the populations too. Some genetic variants (which is what mutations are) become more frequent, some less frequent. Eventually some of them reach 100% in the population, and no more change is possible, at least until another mutation occurs at the same site.