What term is used to describe the following situation? A situation where the movement of the opponent's pieces gets more and more restricted as the attack develops. The opponent can't make moves easily without losing something.
As @Ywapom notes, Zugzwang is often used to the describe the end of this situation, like the Immortal Zugzwang Game where Sämisch suffers this fate against Nimzowitsch.

For the progress itself, it's perfectly normal to use non-chess specific terminology here, e.g. you could say White was slowly getting strangled by Black in the game above.