We all know that to see through something that is opaque to visible light we can use x-rays. For example, you have perhaps had your luggage x-rayed at an airport to be sure it contains nothing dangerous. If x-rays go through solid material, how can you reflect them from a mirror to make an x-ray telescope?
Astronomers make use of a trick that you may have noticed. If you look at a normally unreflective surface at a grazing angle, it reflects a little bit. A polished table surface is a good example. If you look straight down at it, you can barely make out your face, but looking across the surface you can see a relatively good reflection of the distant wall.
To exploit the effect that light reflects better at a "grazing incidence," astronomers design x- ray telescopes very differently than standard optical ones. For example, the Chandra satellite uses "tubular mirrors" to collect and focus the light. The tubes are like paper towel tubes with shiny insides. The light passes down the tube, glancing off the wall and is thereby reflected to a focus where an instrument can detect it. Take a look at the Chandra x-ray telescope satellite web site to see how well this works. The many images there were made taking advantage of this technique.
No one would try to look at the stars using a telescope that was heated to 1000 K and was glowing. The stars' feeble light would be totally swamped by the much brighter light of the telescope itself, making the stars invisible.
Astronomers who want to make infrared observations face a similar problem. Their instruments at room temperature glow so strongly at infrared wavelengths that they overwhelm the signals from the objects they wish to study. How then can you observe at infrared wavelengths? You must cool the telescope. Infrared telescopes therefore use instruments and often mirrors that are cooled with liquid nitrogen or liquid helium. At the low temperatures of these liquids, the telescope's own emission becomes dimmer and allows astronomers to see the objects they really want to see.