The CPU generates a lot of heat while doing its job, and if the CPU can't get rid of the heat, that's very bad for the CPU. (Scientists say the CPU 'fails'. Engineers say the CPU 'dies'. Users say the CPU 'burns up'.)
If you look at the CPU in your computer, you'll see that it's mounted on a big black piece of metal with lots of fins and grooves in it, called a "heat sink". There may also be a little fan blowing air past the heat sink, to help take heat away from it. The thermal grease carries heat from the CPU to the heat sink faster than if they were just bolted together with nothing in between them.
