This is a great question! There are two reasons why we never see matter fall into a black hole. The first reason is because of gravitational time dilation; if we dropped a clock into a black hole, we would see it tick slower and slower. The second reason is because of gravitational redshift; as a luminous object falls into a black hole, the photons emitted, lose energy climbing out of the gravitational field, causing the entire spectrum to be shifted into the infrared, which is outside of the range of visible light that we can detect.
However, both of these effects are only strong and significant very close to the black hole. For a black hole the mass of the Sun, the size of the black hole is only a few kilometers. So for you to be affected by the time dilation effects of the solar-mass black hole (ignoring the gravity!), the black hole would have to be in the same city as you! So black hole are allowed to get very close to each other before time dilation becomes strong. Of course, black holes do get close, and merge even, so at some point we must have very strong time dilation. What happens at this point?
Remember that black hole themselves are not actually solid objects that are crashing into each other. But there is another way we can think of black holes to make sense of them:
Whirlpools are places in the ocean where the current pulls us in toward the center where we can't escape. This is similar to the effect of a black hole. In fact, we can even understand time dilation in this way:
Imagine you and a friend get are in separate boats and the two of you have found a large whirlpool in the ocean. Your friend decides to explore the whirlpool while you stay safely far away. To communicate with you, you friend brought messenger fishes; they release the fish into the ocean and they swim towards your boat, carrying a little message. Of course, as you friend gets deeper and deeper into the whirlpool, the messenger fishes take longer and longer to get out and reach you. Eventually, your friend has fallen deep enough into the whirlpool that all the messenger fishes they release are trapped with them; you stop receiving messages.
The analogy to be made here is that black holes can be thought of as whirlpools in spacetime, and the "messenger fishes" your friend released are the photons (and gravitational waves) that are emitted when objects fall into black holes. That the messages are eventually trapped within the whirlpool represents that the event horizon, the point of no return, has been crossed. Notice how there's no physical surface around the whirlpools that represent this point of no return; with black holes the situation is the same.
So now we can answer the question, what does it look like when two black holes merge? It's similar to two whirlpools merging, We can imagine releasing many messenger fishes to make sense of what is happening in this highly dynamical region of the ocean. Thankfully we don't have to imagine too much, one of our alumni, Will Throwe, did this very calculation! The results of the calculation have been made into a video that you can find by following the link:Two Black Holes Merge into One
Note that the black spheres in the video are not the black holes themselves, but rather their shadows, meaning their appearance when we factor in how they deflect and trap light. Even though there is severe time dilation near the surface of a black hole, the surfaces themselves are very dynamic objects and so we must use simulations instead to answer our questions regarding what things will look like if we had a telescope. Thankfully we have many determined and passionate people working on it!
There might be one lingering question: How do we see black holes merge if I will never see an object dropped into a black hole pass the horizon? How do I reconcile this? One thing to keep is that when we picture dropping a clock into a black hole, we imagine the clock approaching more and more slowly, and the black hole just sitting there, not reacting. From binary black hole mergers we should realize that black holes have lives of their own! As in, the "stationary" black hole that we had in our dropping-a-clock thought experiment it not so stationary after all. The moving clock means we are in a dynamical spacetime, and in a dynamical spacetime the black hole horizon actually changes shape! At some point, when the clock gets close enough, the surface of the black hole "puckers" out slightly and "kisses" the dropped clock as it falls in. This is a more accurate picture than the one in which the surface of the black hole does nothing at all. But to arrive at the full scientific answer and arrive at what the full story is, we have to do simulations like Will did! Hopefully we have more cool videos of simulations in the future!