The sentence is saying that mechanical waves need to move through something (a medium). For example, sound waves travel through air and ocean waves travel through water. Waves displace the medium in order to travel, and after waves travel through a certain part of the medium, the medium returns to the way it was before the wave traveled through it in the first place.
Think of swimming. How do swimmers get through the water? They pull and kick, which displaces the water and propels them forward. But is a pool permanently filled with waves after a swim meet? No, a while after all of the swimmers get out, the pool returns to having a smooth surface. (In this case, the swimmers are the waves and the pool water is the medium.)
Sidenote: Interestingly enough, this doesn't apply to electromagnetic waves (like light), which don't require a medium to travel through. That didn't stop scientists from inventing ether and sticking to it for about a century. 🙂
