It's pretty simple. You can use the known laws of gravitation and the visible distribution of matter in a galaxy to predict how fast stars at the edge of the galaxy should move. But when we do this we find that the stars are moving much faster than expected. So EITHER the formula for gravity is wrong OR there is extra matter there which we can't see, i.e. dark matter. Dark matter is currently the preferred explanation because it also explains a bunch of other, unrelated, phonenomena, but it has as yet never been directly detected or measured.