Why does "deadheading" Petunias help them to flourish?

A lot of flowers do this. I dead head my Roses. What happens, the flower will go to Seed. This stops the Flowering process. When you Dead Head it causes the Plant to produce more Flowers in an attempt to Seed. This easy way to tell it.

Removing the deadheads prevents the plant from using nutrients to try bring the deadheads back to life.

when you deadhead any flower, it redirects the plants energy from producing a seed pod from the spent blossom, which takes lots of energy, to producing more flowers to make more seeds from which doesn't take as much!

What snagel said.

Whenever a flower (and some non-flowering) plants are dead-headed, they are no longer putting their energy into making more flowers. Their energy (and resources) are instead used for growth and continued root establishment.

Actually, pinching the stems back helps a lot too.