Black Holes area region of space-time, which nothing can escape, even light. It is impossible to see a black hole directly because no light can escape from them; they are black. But there are good reasons to think they exist.
When a large star has burnt all its fuel it explodes into a supernova. The stuff that is left collapses down to an extremely dense object known as a neutron star. We know that these objects exist because several have been found using radio telescopes.
If the neutron star is too large, the gravitational forces overwhelm the pressure gradients and collapse cannot be halted. The neutron star continues to shrink until it finally becomes a black hole. This mass limit is only a couple of solar masses that is about twice the mass of our sun and so we should expect at least a few neutron stars to have this mass. (Our sun is not particularly large; in fact it is quite small.)
A supernova occurs in our galaxy once every 300 years, and in neighboring galaxies about 500 neutron stars has been identified. Therefore we are quite confident that there should also be some black holes.
If you were to watch ETA Carin becoming a black hole, you would first see the dramatic and wonderful supernova, in which the superstar forcefully ejects most of its atmosphere very quickly. It's a violent process but quite pretty. After enjoying this for a while, you would see the star itself start to collapse. After watching this for a while, you would think it mighty peculiar that the star seems to be slowing down and getting redder and dimmer as it collapses. These are the effects of the gravitational field getting stronger. Okay, a brief side-note about gravity and light cones