If black holes are invisible how do we know they exist?
True, black holes are invisible. But we can know they are there by the interaction of light on black holes.
First in the late 1790s, John Michell of England and Pierre LaPlace of France hypothesized that there can be a "invisible star". Michell and LaPlace calculated the mass and size which is now said to be event horizon that an object needs in order to have an escape velocity greater than the speed of light & in 1967 John Wheeler, physicist, applied the term black hole to these collapsed objects.
Astronomers comes to know that there is a super-massive black hole in center of elliptical galaxy M87. This is discovered by velocity measurements of a whirlpool of hot gas orbiting the black hole.
Based on the kinetic energy of the material whirling about the center, the black hole is about 3 billion times the mass of our Sun and smaller than our solar system and this how black hole is discovered.
Black holes have 100% Black-body radiation. What that means, is that black holes absorb every single wavelength of all electromagnetic waves and mechanical waves.
Now consider a source of light like Sun. And put a black hole in-front of that source. Surely, enough sun's light will be absorbed by the black hole but some light will curve around the black hole.
By that curved light, we can tell that there is a black hole there. See this GIF below for more accurate understanding of how black holes curves light.
If you do want to see this phenomena more practically, then I suggest you to see a movie called as Interstellar.