The rifts, or trenches, that cut across the mid-ocean ridge prevent the ocean floor from actually growing.
The old ocean floor is dragged down into the trenches by the movement of the mantle, where it is destroyed by the process of subduction.
Subduction occurs when a continental plate meets an ocean plate and the ocean plate is pushed under the continental plate. The part of the ocean plate that is pushed under winds up in Earth’s mantle, where it melts into magma.
This movement of plates is usually accompanied by earthquakes, which are common in ocean trenches. These earthquakes are most likely also responsible for the mountainous quality of the mid-ocean ridge.