Sure, squats strengthen your core, but the core isn't a set of magical muscles that automatically make you faster. Strong quads, hams, and glutes are more important. Squats train your legs/posterior chain/core (and upper body to an extent), and also teach the triple extension [0] which is key in explosiveness. Type I and Type II muscle fibers matter as well

[0] http://myweb.facstaff.wwu.edu/chalmers/PDFs/Triple%20Extensi...