As far as I remember, Einstein just (after briefly mentioning failed experiments to find the ether) postulated c to be constant in any reference frame and then looked what that would result into. For me the most important was that he found symetric equations for the motion of a magnet, where they weren’t before that postulate. In the end it was just his genius to see that if we can’t find that ether that until now (1905) we needed to explain some phenomena, what would happen if there is none. And it turned out that his results offered much simpler explanations, especially symetric ones for the same observations in different frames than before. At the price of some real strange features of space and times of course.