Grifola frondosa is a parasitic fungus - getting its nutrients from the oak roots. Fortunately it is a "good" parasite, not killing its food source but keeping it alive as long as possible in order to maximize its own life. Unfortunately, most of the time the host tree eventually dies, probably from a combination of the Grifola infection and environmental stresses such as drought and wind or because of other fungal issues.

Harvesting the Grifola won't necessarily prolong the tree's life. The parastic infection will remain present underground - the sheepshead or hen of the woods is just the visible fruiting body.