They don’t emit any light, so how do we see them?
There’s more to a black hole than just the event horizon

Whilst matter that falls past the event horizon can no longer be observed, we can still see matter outside the black hole. Hot gas tends to form a bright disk around the black hole, called the accretion disk. This is easily detectable since the radiation emitted from these disks can span the whole electromagnetic spectrum (radio waves to gamma rays).
Quasars are the best objects to observe with this technique. The light from the accretion disk is so bright it outshines the entire galaxy!
Orbits of stars in a galaxy
With simple use of Kepler’s Laws and Newton’s Law of Gravitation, you can calculate the mass of objects from orbital data. We do this in A-level physics a lot. By observing stars, which do emit light, you can determine the mass and relative position of the black hole it’s orbiting!

Of course, I have oversimplified a bit, and let’s not brush off the other challenge that is actually gathering data!
This technique of using orbiting stars was how we detected Sagittarius A*, the supermassive black hole at the heart of our galaxy. The orbits of 90 stars was used to accurately find the black hole.
If you want to learn more about using Kepler’s Laws for binary systems, I have A-level summary notes here.
Merging Black Holes create a lot of noise
Not sound noise, but gravitational noise!
A black hole binary is a system of 2 black holes orbiting each other. It is a highly unstable system, so when they merge together they emit a lot of gravitational waves.
LIGO, a gravitational wave detector, managed to detect waves in 2015 coming from a 29 solar mass and 36 solar mass black hole merger 410 Megaparsecs away. The waves themselves were fairly intense given how immensely weak gravity tends to be, but most importantly the direct observations matched closely the theoretical predictions for the merging of the black holes.
Above is a simulation of 2 black holes merging together. It is one my favourite ones of all time! Notice how the rest of the galaxy wobbles fairly slowly as the black holes merge, but then shakes more violently once the merging is almost complete? This is a gravitational wave, and what LIGO was detecting.
A couple more gravitational waves have been detected since, which is a huge achievement for astrophysicists all over the globe! Gravitational waves are so weak that the equipment used to detect them has absolutely no room for inaccuracies and errors.