Why won`t my whirlpool dryer not spin when I push the start button

Several reasons starting with a popped circuit breaker. Start by testing the outlet, let me know if this is 240 vac before you test.

Look at these links and follow instructions:
http://www.electrical-forensics.com/Dryer/ElectricClothesDryers.html
http://fixitnow.com/wp/2010/10/09/whirlpool-dryer-model-le5800xsw3-wiring-diagram/

You will have to take it apart and test the thermal fuse near the heating element and the heating element. You will be able to see the heating element and you can see if it's burned in two. There are also fuses at the ends of the element that will need testing, too. You will have to clean the drier out thoroughly or it is all for nothing. Every air duct will have to be removed and scraped and washed out to remove baked on lint deposits which is like a concrete lining. This will burn so the manufacturer has installed the safeties to keep from having a house fire. Driers were the number one cause of fires at one time. Look at: http://www.youtube.com/watch?v=MMP7BW1lLs4 to see how. Also: http://www.youtube.com/watch?v=iBiRw-KElC4 ; http://www.youtube.com/watch?v=yFTZGgVisCI
When you finish cleaning the drier and install a new safety or element it is best to also replace the belt, too, because you do not want to have to disassemble it again and if the belt is over two years old you should replace it. The safeties are oval-shaped with a round center part containing the safety with two plug-on terminals for the wires.
Using an ohm meter you put a lead on each terminal of the safety with wires unplugged and if it's good you will have a reading of about zero; if it's bad it will read infinity.
Note that on most of the videos it does not show them washing out the parts, but this is important because it's the only way to really get the parts clean, just don't get motor wet or the capacitor if it has one. Vacuum out the motor. I would move the drier out to your garage to have plenty of space to work. You also want to check your vent and clean it, also. The job is really easier than it sounds. There is no shortcut that will work; it has to be cleaned well. Before you re-assemble it make sure to use white lithium grease (high-temp grease) and grease all bearings. Just vacuuming out the dryer a little does little good. DrTom4444