As @longtresses suggests, there are mixed messages when one looks at the whole spectrum of Buddhism. The Buddha refused to endorse any particular view of the self, either to confirm or deny it’s existence. He did teach that all things are ”anatta”, or no-self, but this teaching turns out to be for more nuanced than the nihilism it seems to imply at first glance.
That all has a bearing on the business of an afterlife. If no personal essence traverses time and space, then even before death, no “person” survives from one moment to the next. That’s obviously not how we conventionally see things, but Buddhism sees this as being an aspect of the truth. In a sense, nothing carries over from one moment to the next, much less from one life to another. There are no “beings”. Only the moment is.
But Buddhism also recognizes that this other level, on which we function as individuals living out our personal lives, is also valid on its own terms. These views seem contradictory, and they may be impossible to reconcile on an intellectual level. Buddhism leads one to an intuitive—not intellectual—understanding in which these two aspects of the truth are seen as two sides of a single coin, each indispensable and inseparable.
When you accept the multi-faceted nature of person-hood in Buddhism, then it becomes less surprising that Tibetan Buddhists keep looking for their reborn lamas or that the Buddha kept referring to his past lives as this or that animal, all while acknowledging that ultimately there is nothing to be reborn.
Various analogies have been used to clarify this. A being can be thought of as a wave on the open ocean. It certainly has the feel of being an entity, distinguishable from other waves there on the surface. It looks as though something is moving along from there to there. But in fact, nothing at all travels from there to there. Energy—the fundamental fabric of the universe—is just lifting and lowering the skin of the ocean, now here, now there. The wave isn’t a thing that can be isolated, tagged, picked up and moved to another part of the ocean. It is the ocean in one of its various manifestations. And yet, we’re not entirely wrong to speak of the wave as if it were a thing, traveling from there to there. Both are true at the same time.
Beings, including us, are dynamic processes propagating across the ocean of whatever all this is. There is the great overall Process of the moment-by-moment unfolding of the universe, but it’s possible to focus more on the minutia, as we typically do, and see it as a collection of individual processes, beings within Being. From the larger perspective, the eventuality of what becomes of all those little processes is not a great mystery: they were just wrinkles in the great unfolding, never their own private domains in the first place, and the energy of those smaller processes just goes on to carry the unfolding forward. “Death” is a non-event.
It’s only because we are so concerned with the fate of our own little process, and that of those we love, that we fret about what becomes of the “person” that we’ve come to associate with each process. There’s nothing wrong with that. It’s how we are. The world would be less beautiful without it. But it’s not the whole truth.