String theory is a mathematical framework proposed to unify the strange phenomena observed at very small length scales (quantum physics) with Einstein's theory of general relativity. The mathematics of both quantum physics and general relativity make strange predictions that have not, as yet, proven false.
The fact that string theory predicts 10 dimensions is also born of mathematical formalism; the biggest issue is that the existence of 10 dimensions is very hard to prove or disprove.
To address your question, we will assume that the 10 dimensions predicted by string theory are correct, but we should begin by assessing the relativistic physics relevant to the concept of time travel. In summary, time moves more slowly for objects moving at high speed. For objects moving at the speed of light, time stops. As such, if you were to take a lap from the earth to the sun and back at the speed of light with a stopwatch, you would perceive that journey as instantaneous and your stopwatch would read 0 minutes and 0 seconds.
But from my perspective, the journey would take you 16 minutes, and when you returned, my stopwatch would read 16 minutes. Effectively, you traveled 16 minutes forward in time. You may next ask how you can travel backward in time. If you think about graphing the "speed" of time as a function of the speed of the object experiencing it, you could draw x-y axes and (for discussion purposes) a simple line with a negative slope, intersecting the x axis where the object's speed equals the speed of light. Time would then have NEGATIVE "speed" at object speeds FASTER than the speed of light. This implies that to travel backward in time, you would have to travel faster than the speed of light. So far, no physical object is known to be able to do that. The fastest objects measured on Earth have been subatomic particles in the large hadron collider at CERN in Switzerland, at 99.99% the speed of light.
If 10 dimensions do exist, a type of time travel could be possible by means of higher dimensional transport. The key concept is explained in the movie Interstellar (watch the scene at here ), where the piece of paper is a 2-dimensional world, and travel within those two dimensions has to take place in a path that can be drawn across the page. In our higher-dimensional 3-D world, we are free to fold the paper out of plane and travel between two points on it without occupying the 2-D space between them. If we extend our thinking to a spatial dimension beyond our 3-D spatial world, "folding" through a higher dimension might enable us to arrive at a different point in space. In fact, we already do that: think of a 3-D spatial world without time, meaning the whole 3-D world is just a snapshot of one moment in time. If we want to travel to another point in that 3-D world, we actually have to spend time doing that. In that sense, time is our 4th dimension, that allows us to come to a different spot in 3-D space. Thus to travel in 2-D space without time, you need 3-D space, and to travel in 3-D space, you cannot travel without time. By extension, to travel arbitrarily in time, your vehicle must be in a higher dimension.
But how can we think about what that would feel like? Our experience of 3 spatial dimensions and how there could be higher spatial dimensions that we cannot perceive is well-articulated by the "bug-on-a-wire" argument, found (go to page 5 and read the paragraph under "Introduction" at here ). The filmmakers of Interstellar play with the concept of time as a dimension in the end of the movie (watch at here ). Here, the time dimension is portrayed as spatial, and Matthew McConaughey travels around in space to visit different moments in his daughter's life through the lens of her bookshelf. The premise is that Matthew McConaughey has happened upon beings which live in a higher-dimensional world and experience the dimension we know as "time" differently; they demonstrate this to MM in a way he can understand, using 3-D space. Extending this way of thinking, the way that higher dimensions (i.e. 5-10) can be experienced could resemble space, time, or something entirely different. So for what means higher-dimensional means of transport might actually look like, your guess is as good as mine.