Dark matter is a pretty reasonable guess to explain a very clear observation.

You’ve seen animations of the solar system, right? The inner planets are wizzing around really fast, but the outer planets are going quite slow. A galaxy should work the same way. But when you measure the speed of stars at various distances (using red shift and blue shift) the speed is a lot more uniform.

So, something is missing. Maybe red shift/blue shift doesn’t work the way we think it does. Maybe gravity doesn’t work the way we think it does. Or maybe there’s a lot of extra mass we can’t see.

For a long time (and mostly still today) people thought the most likely explanation was missing mass. We know there are particles we haven’t discovered. The discovery of the neutrino happened just like this: an experiment got a strange result, and someone proposed a new particle to explain it.

But with the neutrino, they came up with a clever way to test the hypothesis, and when they did, they found neutrinos.

Actually, I suppose a similar thing has been done with dark matter. Given the hypothesis called dark matter, they make a prediction: light should bend around galaxies a lot more than it would otherwise. They set out to find examples and it took the resolution of the HST to finally find any. “Gravitational lenses” they’re called.

But unlike with the neutrino, this doesn’t give us any insight into what dark matter actually is. So, people are still open to other possibilities. MOND is one.

Brane theory is the latest one I've heard.