To "erode" means "to grind away, to wash away, to wear down." It often means slow damage.
"Profound" means "deep."
A "profound erosion of middle-class jobs" means that year by year, middle-class jobs have been slowly lost (erosion) and that the total loss is large (profound erosion).
To "reverse" means to "make something move in the opposite direction," like putting a car into reverse gear.
He is saying that even though Americans are finding work, the underlying problem continues and the improvement may be temporary. It will take a long time to reverse the fundamental causes of the problem.