how do we know that c is the limit? and why is it the limit?
There are two common logical frameworks you can use. (1) You can assume that there is a particular, special speed c that is the same in all frames of reference, plus some assumptions about symmetry, esp. that all frames of reference are equally valid. You then conclude that the Lorentz transformation holds, and that time is not absolute. (2) You can make the same assumptions about symmetry, and also take it as an empirically established fact that time is not absolute (e.g., based on the results of the Hafele-Keating experiment). You then conclude that the Lorentz transformation holds, and there is a particular, special speed c that is the same in all frames of reference.
Nothing so far requires that c be associated with the speed of light. In general, it's an archaic mistake to think of the c in relativity as being the speed of light. It's fundamentally a property of spacetime, not a property of light. However, if photons are massless, then one can also prove that they move at c. Also, the Michelson-Morley experiment seems to show that the speed of light is the same in all frames of reference, which requires that it equal c.
Relativity does not in fact say that c is the maximum speed. However, it does put tight constraints on faster-than-light (FTL) motion. In particular, it says that any mechanism for FTL transmission of information is also a mechanism for transmitting information from the future into the past. This would violate causality. This is the most general thing you can say about FTL in relativity, just that it's inconsistent with causality. In more specific cases, you can say more. For instance, in the case of a material object, the energy required to accelerate it to c would be infinite.
So in summary, if you want FTL, you have to give up causality, or you have to find something wrong with both the Michelson-Morley experiment and the Hafele-Keating experiment (as well as many others).
A good book on this topic is Time Travel in Einstein's Universe, by Gott.
Some people have seriously investigated the possibility that causality is not generally necessary in order to have reasonable, self-consistent laws of physics. Some interesting papers:
A very good answer, but I would put this sentence differently:
Actually, it's the principle of relativity (in SR) that is at odds with FTL travel and causality. If FTL had a special frame in which it works, there'd be no problems with the experiments. So, following JesseM: You can have exactly two of the following: FTL travel, relativity, causality.
This is a very nice formulation.
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