Image: Space Telescope Science Intitute
Black holes, by definition, are regions from which light cannot escape. Thus, any radiation observed from outside must come from outside black holes.
There are two mechanisms by which this can happen. The first, Hawking radiation, comes from the behavior of quantum fields in the presence of strong gravity. This type of radiation is only important for very small black holes, and to detect it, astronomers look for a signature of light from matter that heats up as it falls into the black hole.
To understand why objects get hot as they fall into a black hole, we must first analyze the forces that act on them as they fall; after all, we are not accustomed to thinking of things heated by gravity!
Although the force of gravity is usually described as being directed towards the center of a black hole, the situation is actually a bit more subtle. Imagine yourself falling feet first towards a black hole. Your feet are a bit closer to the black hole than your head is, so they're pulled a little more by gravity--and you get stretched a bit. In addition, your left and right hands don't fall along parallel lines, but rather both fall towards the center of the black hole. As a result, you're squeezed a bit as well.
The upshot of all this is that gravity, in addition to pulling things in, stretches and compresses them too. And this squeezing causes heating, leading to the emission of radiation long before the falling matter actually enters the black hole. It is this type of radiation that scientists look for when they search for black holes.
John Gaustad, a professor in the department of astronomy and physics at Swarthmore College, adds the following.
Indeed, light cannot escape from within a black hole, but the matter falling into a black hole can get pretty hot before it falls in. This heating is due to the fact that matter accelerates near black holes. As a result, matter emits lots of light and other radiation as it falls into a black hole.
Among the best evidence for the existence of a black hole is a source of x-rays coming from a binary star system in which no star companion is observed. In other words, there are visible stars whose orbital motion tells us that they have an invisible partner. An invisible companion emitting x-rays is sometimes a neutron star, but neutron stars cannot exist above a mass of about two to three times that of the sun. So if analysis shows that the invisible x-ray-emitting companion has an even greater mass, we know it must be a black hole.