There is nothing in science that prevents time travel explicitly, but time travel poses a real problem to science and it isn't the problem you may think. The problem with time travel is the possibility of a paradox, and the common one most people think of is the Grandfather Paradox. You travel back in time and kill your Grandfather. Since you killed your Grandfather, you would not exist to go back in time to kill your Grandfather--hence the paradox.
There is a more subtle one as well. On a dark and stormy night, you hear a knock at the door. When you open the door a man in a strange suit hands you a worn book. The man looks strangely familiar, but you can't quite place him. He says to you, "In twenty years, the technology will be available to build the machine in the book. Build it, go back in time, and give yourself the book." The man then turns and walks away. You look through the book and discover it is the plan for building a time machine. This paradox is actually worse than the Grandfather paradox, although t doesn't seem so at first glance. Think about it for a moment and ask yourself the question: who originally wrote the book? Where did it come from?
The problem is the same in both of these paradoxes, in fact in every paradox. Paradoxes violate the Law of Conservation of Information. In the Grandfather paradox, information is being erased from the universe. In the second paradox, information has been added to the universe and both violate the Law of Conservation of Information.
Conservation of information is a corollary of the Law of Conservation of Energy and is a consequence of Einstein's famous E = MC^2 formula. The formula states that mass and energy are equivalent, they are just different forms of the same thing. The Conservation of Energy states that energy cannot be created or destroyed, it can only change its form.
Mass is energy, but it is energy with information attached to it. In fact, all energy is a form of information since mass and energy describe the universe around us. An atom has some information associated with it, and when you split that atom, that information must be conserved the same as energy is conserved. The information of the two components equals the information of the original component.
Time paradoxes violate this information conservation. In the case of the Grandfather paradox, information is being removed from the universe. Information cannot be removed from the universe, you can only change its form. In the second paradox, you are adding information to the universe, information that doesn't have a source. Both of these conditions violate information conservation but are possible if time travel is true. The paradox is that the Law of Conservation of Information is violated, something that simply you cannot do under science as we know it. If the conservation of information were possible, then physics is wrong and the whole scientific framework that we use to understand the universe just crashes. However, the laws of science have been proven over and over so it is the paradox that cannot happen not that the laws of science fail.
It is strange that nothing in science prevents time travel, probably because the notion of time in science is actually quite vague. It is really hard to define exactly what time is and if it actually exists. Yes, we have space-time, but that concerns the passage of time in a frame of reference as indicated by a clock. Clocks actually don't measure time. Clocks are simply chronometers that operate at an agreed-upon frequency that help us measure the world around us and synchronize activities around the solar system. Clocks are just fancy metronomes, that tick at a certain constant rate (in a perfectly flat portion of space-time not influenced by any measurable gravity). The agreed-upon rate is dictated by super-accurate atomic clocks--which also don't measure time.
Time paradoxes probably indicate that time travel isn't possible, given that the possibility of a paradox, even if that possibility is very rare. Maybe you can't change the past at all. For example, it may be impossible to kill your Grandfather even if you tried. The second paradox may not be possible either, but we simply don't know, and that is a problem. The very possibility, as improbable as it may be, may indicate that time travel will forever remain in the realms of science fiction.
