An intercooler is a heat exchanger. That means there are two or more fluids that don't physically touch each other but a transfer heat or energy takes place between them. At wide open throttle and full boost the hot compressed air coming from a turbocharger is probably between 250 and 350 deg F depending on the particular turbo, boost pressure, outside air temperature, etc..

We want to cool it down, which reduces its volume so we can pack more air molecules into the cylinders and reduce the engine's likelihood of detonation.
How does an intercooler work? Hot air from the turbo flows through tubes inside the intercooler.