Okay, an intercooler is just a small radiator, just like what cools the engine itself. There’s a “manifold”, or “plenum” at each end, connected by a row of a bunch of small tubes (to give greater surface area) through which the water flows.
Water flows through the tube, the air passing over it carries off some of the heat, cooling it. The cooler water leaving the other end then goes on to cool the engine, and it carries that heat back to the radiator to start it all over again.
Now, take that same radiator- basically-and put a water-tight duct around the tubes, so you can run water through where the air used to go. Then just run your turbo-air through where the water used to go.
It really is just that simple.
The act of compressing the air- as in the turbocharger compressor stage- heats it up, and that heat reduces the intake charge density (which is exactly opposite to what the turbo is trying to do in the first place.)
The intercooler takes away that excess heat, making the charge air denser, providing more power.
As for letting the car idle for a while, the “cooling down” has to do with the impeller bearings in the turbo housing. The impeller is force-fed pressurized oil that both lubricates the shaft (which can be turning 100,000 rpm at full scream) as well as cools it.
If you blew right in from the freeway at sustained 80MPH speed, power-slid into the driveway and then immediately shut the car off, the turbo would still be smokin’ hot, but has now lost that cooling oil in the bearings.
The oil that’s in there basically cooks- it breaks down and carbonizes from the heat (1,200 degrees F isn’t impossible.) And naturally, this carbon “coke” is not good for either the turbo or engine- when you go to start it up next time, the bearings are scored by the debris, and then flushed into the engine.
Do this for a couple of weeks during your commute and the bearings get rapidly destroyed- the turbo begins to leak oil into the exhaust and pressure stages, the impeller won’t spin as well, reducing power, and could simply seize up. Impeller seizures tend to be catastrophic failures, by the way. 