Ok, here's how thermal paste works. The bottom of the heatsinc, and the top of the die, are not MADE for eachother. They have imperforations in the surfaces, which cause microscopic holes and grooves. If there is not a complete seal between the heatsinc and the CPU die, then there will not be effective cooling because the heat from the die can't get to the heatsinc.
Thermal paste is like silver paste, or zinc paste, or whatever. It conducts heat. So, when you put thermal paste ontop of the heatsinc, it can bond with the CPU die almost perfectly...filling in all the microscopic holes and grooves. So now, the heatsinc has almost 100% contact with the CPU die.
Thermal paste is very cheap, and easy to apply. To remove the old stuff, carefully warm it up with a hairdryer, and then use rubbing alcohol or acetone (nail polish remover) on cotton swabs to remove it. Make sure all residue and any cotton pieces are gone before you re-apply it.
Put a small blob of thermal paste in the center of the bottom of the heatsinc. Take your finger and spread it out in a thin but complete layer, big enough to cover the die and that's pretty much it.
EDIT: By the way...
CPU's get VERY hot VERY fast. The second you switch on your computer, your CPU is already hot. If you were to take off your HSF and run the computer without it, the chip would heat up so fast that it would be fried before you left the BIOS screen.
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