We don’t know that light speed travel is impossible, but our current understanding of physics certainly doesn’t have any realistic method for doing so. Relativity has a lot to say about what happens when you accelerate a massive (as opposed to mass-less) body to near the speed of light, and the short story is that plain acceleration is not going to work.
Back in 1994 Miguel Alcubierre proposed using a device to alter the geometry of spacetime around a ship, allowing it to move slowly through warped spacetime to achieve a much higher speed relative to objects outside of the warp. While there’s a lot of hype around Alcubierre Drives the reality is that right now we have no good reason to even believe that it is possible to build without a major revision of certain parts of physics since the current theoretical models rely on finding a substance with negative mass – which current physics does not allow for.
Infeasible though it seems at the moment there is a possibility that the Alcubierre Metric can be achieved through other means, or that we may improve our physics understanding to resolve the current issue.
Or we might resolve the problem with Einstein-Rosen Bridges (wormholes), or prove the existence of tachyons and leverage those for FTL travel, or find some other way to cheat the apparent speed limit. We can speculate, but there’s currently no real science that offers FTL communication or travel.
It may be that the speed of light is really the absolute limit that cannot be exceeded, but at the moment we can only suspect that to be the case.