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.

Reverse means "turn around" or "go back" here. Profound means "deep" or "substantial" and erosion means "wearing away" or "loss". Within this sentence a "profound erosion of middle-class jobs", it means that there are not a lot of middle-class jobs left and it will take a lot to get them back.