Thatch is a a mostly hidden but sometimes visible layer of dead grass, leaves, stems that accumulates near grass roots over time. The more thatch that is present, the "squishier" the lawn will feel when you walk over it.
Dethatching or power-raking removes this buildup of thatch layer which better allows water, oxygen, light to reach grass roots and soil directly underneath and keep the lawn healthy.
Dethatch should be performed once every 2-3 years regardless of whether you bag clippings and perform fall cleanup of leaves. Grass clippings from normal cuts and residual leaves both contribute the an increase in the thickness of the thatch layer.
Dethatching is performed using a power rake and then the thatch is bagged using a regular mower. It's typically a several hour process and the price ranges from $50 - $75 depending on lawn size.