
Firstly, it's a fact that when the air enters the engine from the air intake it must be at a sub-sonic speed. Inside the air intake if the F-14 there are a few hydraulic-propelled "plates" that compress the air while the plane flies at super-sonic speed (it's deployed automatically), and therefore the air slows down before it reaches the engine. This system is equivalent to the moving intake of the F-15.
So, in that scene they fly in a sub-sonic speed, and suddenly they cross another jet's trails. That fast-moving air reaches the engines too fast (that system wasn't deployed since they flew in a subsonic speed), and the engine stalled. As a result they entered a falt spin, which is hard to recover from.