How can astronomers "see" dark matter?
They can't, and it might not even be there.
Astronomers observed nearby galaxies and tried to predict their movements based on the presumed mass of observable phenomena. The difference between the predicted movements and the actual ones implied that the galaxies were a lot more massive than they ought to be, and "dark matter" is a name for the unobservable mass that would account for the difference.
Other scientists claim that there is no such thing as dark matter, that gravity just works differently than current models say it does, and we should adjust our ideas about gravity to account for the difference instead of hypothesizing a vast, unobservable phenomenon to explain it.