Why would I add worms to my compost pile?

The compost pile gets hot. My is usually 120-140 degrees and I'm trying to get it hotter. Wouldn't worms just die as fast I I can put them in? Or, after the pile gets over its initial hotness will the worms just start coming up from the ground and finish what the heat started? Also, are all worms created equal or do I need to use a specific kind of worm? I saw videos that specified "compost worms". Whats that?

You would not but many times you can find them at the bottom of the compost pile and the top of the soil.  A good place to collect them.

Adding them to the compost pile will while it is cooking will kill them but after it is all good.

Generally people will do vermicompost or do traditional compost.  

Vermicompost 1

Vermicompost 2

Compost, extracts and teas

Compost worms are Red Wiggler worms you can find for sale as fishing bait, they eat vegetable matter directly unlike most earth worms, these worms tend to live in the upper foot of soils.

You don't need to add worms to a compost heap, they will come when the heap is read for them to do their work. I consider worms as the indicator of the heap being almost finished and ready to use in the garden.