It's really pretty simple though.
Electricity is when an electron passes from one atom/molecule to another. This is electrical conduction. Static electricity is a perfect example. You preferably want a thermal paste that doesn't conduct electricity well because the last thing a CPU needs is too much electricity.
Likewise, thermal conduction is when energy stored as heat passes from one atom/molecule to another. Such as if you leave a penny out in the sun and go to pick it up, it burns you. Why? Because it is conducting it's stored heat into your skin.
Thermal conductivity is the whole point of that heat sink. It tries to take the excess heat from the CPU chip and pull it away into the heat sink. Then the air flow (usually from a fan on the heat sink) conducts the heat into the air where it gets carried away, leaving the CPU at a cooler temperature than it used to be.
So the more conductive you can make the heat sink, the more heat gets pulled away from your CPU.
Thermal paste and other such things are made to improve this thermal conductivity between the CPU and the heat sink. Because the metal heat sink and the silicon CPU have microscopic sized warps in them, it isn't in a perfect 100% connection to the CPU. And that results in a less than ideal heat transfer.
What thermal paste does is it fills in those tiny warps to give the heat sink a much better connection to the CPU. This results in more heat being transered away from the CPU, which gives you a cooler chip.
If you don't use something to fill in these tiny gaps, the chip will run hotter. In some cases with very hot chips like the T-Birds, this can result in bad things happening.
So it's always a good idea to use something like a thermal paste. It's just safer that way.
The ideal properties of a thermal paste are that it can melt and/or squish to fill in those little holes, it can conduct heat easily, and it doesn't conduct electricity easily. Arctic Silver is a good example of this. The grease in it doesn't conduct electricity well, but it doesn't impede heat conduction much. And silver is an excellent conductor of heat.
And, as a side note, if you use too much paste it will actually end up blocking the connection to the chip instead of improving it because grease isn't as good at conducting heat as metal is, so you want as much of a connection with the actual heat sink itself as possible. You really only want just enough paste to fill in those microscopic gaps but not to block any connection to the heat sink.
And if you use too much it can melt and drip and run all over, which is icky.
- Sanity is purely based on point-of-view.