Here is how time travel is understood by today’s established science, namely general relativity. To simplify matters, I’ll only talk about single particles time travelling, not sentinent beings: this doesn’t mean there is any fundamental difference, but it makes the whole thing sound much more plausible.
Consider a simplistic model of time travel, described in chapter eight of Science of Discworld III by Terry Pratchett et al. if nowhere else. First, picture the history of an ordinary world on a sheet of paper, with (one dimension of) space running horizontally and time unfolding vertically, from bottom to top. Points of such a picture correspond to events happening in different places at different times. The life of a point particle is represented by a line. Energy conservation means that one can draw how energy flows on that picture, from one point now to another a second later, without any sources or sinks.
I will pause here to remark that this picture has no notion of the present turning into the future: the state of the world “now” is just the points on a correspoinding horizontal section of its history. If you can understand that your idea of a horizontal section can be different from that of someone moving relative to you, you understand the crucial point of special relativity. Basically, all that remains is to find out how exactly it is different.
All you need for time travel is to roll up your sheet of paper into a horizontal cylinder. If the top and the bottom of the original sheet match up, there is no discontinuity anywhere, and the result is as consistent with the laws of physics as the original was. In fact, locally everything looks exactly the same: you can’t tell if you are on a cylinder or on a plane by exploring a small region of spacetime. Globally, however, we see that time has become periodic1. The line representing the life of a particle will necessarily return to the same point again after making some number of turns. Move a horizontal section (a “now”) along the cylinder and you’ll see that the particle turns into a past instance of itself after you make several turns. (This is what’s less realistic for a complex system than for a single particle.) This is not Groundhog Day where everything stays the same except you: in a very real sense, the previous morning is the same thing as the next one, and you can’t help repeating yourself. There is much less hope in a real physical time loop than in a fictional one.
On a cylinder as on a circle, energy can flow without being created or destroyed. Energy conservation need not be violated. Moreover, energy always flows from the past to the future. It’s just that after flowing into the future for some amount of time, it finds itself back in the past, and so does everything else.
Feynman’s idea of antimatter as matter going back in time is an intuitive description of a completely unrelated and rather technical point in quantum field theory. Consider drawing a circle on a plane (not a cylinder) and moving a horizontal section from bottom to top to watch the story unfold. You’ll see two particles appear out of nowhere, move away from one another, then turn around, meet each other again and disappear. If you complain about energy conservation here, you’ll be right: in fact this process, called pair creation and annihilation, needs energy at the beginning and releases energy at the end—usually, but not necessarily, in the form of gamma radiation (a fancy word for really, really blue light).
In the periodic universe, everybody who lives long enough is a time traveller, and must relive the same thing time and time again (although they probably wouldn’t experience it that way2). However, if you let the shape of your sheet of paper become a bit more complex—a plane with a mug handle attached, for example,—you can imagine a line—a life–that goes around two times around the handle and then continues into the future. But whatever the universe, a time traveller can’t change the past, because time is just another direction on a sheet of paper! The only thing he can do is experience the same moment two times from different points in space.
Finally, just for fun, consider curling up not time but space—a vertical cylinder. Now if you go far enough to the left, you find yourself arriving from the right. The world is finite, but there are no boundaries anywhere. This is what people mean when they talk about a finite universe.
1 The cognoscenti should note that this periodic Minkowski time is not the same as periodic Euclidean time used to turn quantum mechanical evolution into partition functions of statistical mechanics.
2 I don’t think there can be a thermodynamic arrow of time in this world, so it wouldn’t bear any resemblance to the real one, but I’m not aware of any descriptions in the literature. Edits welcome.