"Why do we use ______ rocket fuel?" is complicated.
I recently read John Clark's Ignition!: An informal history of liquid rocket propellants, which I highly recommend. It's exactly what the title implies, and covers the history of fuel development, touching lightly on the pre-WW2 era, then hitting every major development through the sixties (i.e. it covers the time when most of the major development was done).
Before describing further, I'll mention that Elon Musk apparently puts a lot of stock in this book in particular. So, whatever Ignition! says is likely to have had a large influence on the fuels SpaceX uses. However, this influence is likely somewhat perverse: Clark spends about 180 pages gleefully recounting all the amusing ways people blew themselves up, or poisoned themselves, etc. with rocket fuels, while hunting for hypergolicity, temperature range, and an oxidizer that wouldn't spontaneously turn into green slime.
Instead, SpaceX seems to have read ignition as a cautionary tale, and went with liquid methane and LOX. This is most notable for being one of Tsiolkov's original suggested rocket fuels, way back at the dawn of rocketry. Liquid gas handling isn't exactly trivial, but it's way, way easier than most of the other options. In fact, Clark makes this exact point in his chapter on LOX and FLOX*, though he immediately traces the evolution of the most dangerous version of it (slushy liquid hydrogen and freon-ated liquid ozone!).
Overall, I suspect the folks at SpaceX wanted very much to avoid the excitement of modern rocket fuels, and methane was cheap, easy to work with, and had a minimum of exciting environmental impacts. Kerosene is the more usual default, but once you're handling cryonics anyway, you may as well go with a liquid-gas fuel.
* Clark says very little on Methane-LOX systems, but does mention someone's bright idea to make them into a monopropellant: "His idea was to set up a liquid oxygen plant alongside a natural gas well, tank up your ICBM on the spot, and push the button." Clark's horror at the idea of using a LOX-fuel mix as anything other than a bomb comes through clearly.