As for why it’s generally believed to be impossible, the classic example is the “grandfather paradox”. Suppose that I invent a time machine, and go back in time to before my parents were born. I then kill one or more of my grandparents. Now, this means that my parents were never born, and therefore neither was I. But if I was never born, then who was the guy who went back and killed those folks? And if nobody did, then I do exist after all, so why can’t I do it?
Related to this is the problem of djinns, or effects without external causes. Suppose, for instance, that I’m a little more peaceful in my intentions, and go back to visit Bethoven to discuss his music with him. Of course, I bring along a copy of his famous fifth symphony. But I go back to when he was only still working on the fourth. Well, he takes a look at the sheet music I show him, and he thinks that it’s great. I give him permission to publish it as his own, and a few years later, he does. So where did that music come from?
In physics, the only known problem with time travel appears to be the amount of energy required. It’s not that it would require too much, but that it would require too little: Specifically, less than zero. I’ve seen schemes which would only require negative a few grams or so, so we’re not talking astronomical amounts, here. Nor is the time of creation of the device necessarily a factor: A Thorne wormhole time machine could only take you back to when it was first constructed, but an Alcubierre warp field, which is theoretically no more impossible than a Thorne wormhole, would not suffer from this restriction.
Finally, I should add that there are a few possibilities which could theoretically permit time travel, but which couldn’t ever be built, even theoretically. A Gödel universe, for instance (one whose spacetime has a net angular momentum) would contain closed timelike curves, as would a universe containing a maximally-extended extreme Riser-Nordstrom black hole. But in order for either of these things to exist, they need to exist from the very beginning of the universe: Neither can have a beginning point (or an ending point) in time. So it’s conceivable that we might get lucky and someday find such an object, but if we don’t, there’s nothing we can do about it.