Interesting notes from NYU-Poly on the philosophy or relativity. I highly suggest looking at the second set of notes, discussing why relativity was more or less inevitable.
Really, physicists tried everything else first. As weird as relativity is, everything else was either weirder or experimentally demonstrated to be incorrect. It is really the only game in town.
We won't spend a lot of time on the philosophy of relativity and its postulates, as interesting as it is. Our approach will be to accept it, because it agrees with all experiments to date, and see what we can do with it. This approach is sometimes known as "shut up and calculate" -- we do what works, and the second something else works better, we do that.*
Later in the course, relativity will come up again and again ... so if you aren't quite satisfied with the necessity of special relativity by the time we finish it, you can think of it as a useful tool we'll need to have ready for later.
*There are a couple of 400-level philosophy courses that will take precisely the opposite approach, if you are more interested in the why than the how. Nothing wrong with either approach.