We see their effects.
That’s basically how we observe most of the stuff in our universe. Up until a few years ago when we managed to take an actual image of a black hole, astronomers, physicists, and mathematicians would observe the sky (not just with their eye but a bunch of tools), and use fancy-ass math prove, discover, and observe. They noticed that some things “should” be there, but when they pointed their telescopes, they saw nothing. That’s what led people to believe that there “must” be something there affecting the other celestial bodies around them. Through more math and tools, they eventually proved black holes’ existence.
I omitted names of who discovered, or who proposed first coz it’s a bit confusing and I’m not very knowledgeable in that aspect.
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