The relationship between the speed of light and the possibility of ‘meaningful’ time travel (i.e. travelling through more than a few milliseconds of time) is based on an extrapolation of the observed effects of time dilation at sub light-speed velocities.
Time dilates more at greater relative velocities. That is to say, the faster you go, the greater the difference in time that you experience and an observer experiences. This is helpful for the time traveller who, if travelling at a high enough relative velocity, can experience a decreasing amount of time to jump forwards increasing amounts of time; travelling faster means a greater efficiency in jumping forwards in time, and it follows that by travelling at the fastest speed possible – the speed of light, then time travel using time dilation as the mechanic can be achieved most efficiently.
In fact, at the speed of light, the time traveller would experience an infinitesimally small amount of time, and would be able to jump forwards an infinitesimally large amount of time forwards. In other words, the time traveller could instantly move from his own to to any point of time in the future.
Exceeding the speed of light is (currently) not possible, so any consideration as to what might happen if is was is nearly pure conjecture. One point of view, interesting to those interested in time travel, is the extrapolation of the above sequence of events, and it goes like this: at high relative velocities, time slows down. At the speed of light, time is reduced to zero (i.e. instantaneous). It follows (allegedly…) that exceeding the speed of light causes experienced time to reverse – the traveller would arrive before he left, i.e. he has travelled backwards in time.
This extrapolation of experienced time with relative velocity is perhaps a little similar to the extrapolation of decreasing temperature with decreasing volume. Extend the line far enough, and you find that zero volume is found at a given temperature (“absolute zero”, -273 degrees Celsius). Absolute zero is analogous to light speed in that it marks a physical limit. It is not possible to go colder than absolute zero (if you did, there would be negative volume), just as you cannot go faster than the speed of light (if you did, time would reverse).
In summary, the speed of light is important in time travel because by attaining it, instantaneous time travel into the future is made possible. By exceeding the speed of light, time travelling into the past is also possible.