Satellites are isolated systems, the only way for it to transfer body heat to outer space is thermal radiation. There are solar panels, so there is continuous energy flow to inner system. No airflow to transfer the accumulated heat outer space easily. What kind of cooling system are being used in satellites?
Typically, satellites use radiative cooling to maintain thermal equilibrium at a desired temperature.
How they do this depends greatly on the specifics of the satellite's orbit around Earth. For instance, sun-synchronous satellites typically always have one side in sunlight and one side in darkness. These are particularly easy to keep cool because you can apply a white coating to the Sunward side and and black coating to the dark side. The white coating has a low value for radiative absorption while the black coating has a high value for radiative emission. This means it can absorb as little light as possible while emitting more thermal radiation.
Different types of satellites have different strategies for cooling, but in general, cooling is achieved by applying functional coatings to the spacecraft that lower or raise the absorptivity/emissivity/reflectivity of its different surfaces. While designing a satellite, the space engineers perform thermal analyses and lots of calculations to determine which surfaces need to have what absorption values in order for the satellite to maintain the desired temperature.
It's hard for me to be more specific than this. But this is the reason any good space engineer knows how to find a coating with the desired absorptivity/emissivity values within a day or two.
As an example, the International Space Station (ISS) has external thermal radiators. They looks similar to solar panels, but instead of pointing the flat side towards the sun, they point towards empty space. An ammonia loop carries heat from various parts of the space station to the radiators.
This is a picture of a radiator: (source)
- External Active Thermal Control System on Wikipedia
The satellite itself can do with radiative cooling but some instruments on board, e.g., IR sensors, require temperatures as low as than 4 K for which Helium dewars are used. Bolometers require even lower temperatures (in the mK range).
A good summary is available here.