There are 2 load-sharing pieces of cartilage cushion between the femur and tibia, one that sits on the medial side (the medial meniscus) and the one that sits on the lateral side (the lateral meniscus). Both meniscus are anchored to the flatter-shaped tibia to allow the curved femur to slide on top.
This cartilage tissue can tear. These tears can either be traumatic, as would occur in younger people performing athletic events, or degenerative, as would occur a-traumatically in people nearing 40 years and older.
Since most of the meniscus does not have nerve fibers, the tear itself does not cause symptoms. However, the torn tissue getting caught between the two bones (femur and tibia) can. The torn fragment being caught can pull on the joint capsule (which has nerve endings), also causing symptoms.
Fortunately, as long as you maintain your ability to fully extend your knee, these tears can be treated with exercise. The exercises in these videos will not worsen the tear or cause further knee damage, but will encourage the symptoms to resolve.