I did not photograph Biggie doing movements at the knee for some reason, womp womp. Here's some more pictures of me in my sweats and crocs!
What movements happen at this joint?
Flexion is when the angle between the tibia and the femur is decreased by moving them toward each other, and extension is when the angle is increased by moving them away from each other. Of course your knee is backwards, so your lower leg goes behind you to flex and comes back forward to extend.
What muscles make these movements happen?
Best if you take this drawing of semimembranosus and semitendinosus with a grain of salt, I think actually ST is on top of SM; it's hard to tell from the pictures in the book. All I can vouch for is that biceps femoris is the lateral hamstring and ST and SM are the medial hamstrings.
By the same token whereas your hip flexors are anterior muscles, your knee flexors are posterior muscles and vice versa for extensors. As a matter of fact, muscles of the thigh are grouped into compartments; the posterior compartment contains the knee flexors, the anterior compartment contains the knee extensors, and the adductor compartment contains the hip adductors, one of which assists in flexing the knee. Anyway without further ado, your knee flexors in the posterior compartment are your biceps femoris, semimembranosus and semitendinosus, or hamstrings, and also your gracilis from the adductor compartment, and also your popliteus, a small muscle behind your knee that "unlocks" your knee from extension, and also your gastrocnemius (not pictured) or calf muscle, and your knee extensors in the anterior compartment are your rectus femoris, vastus medialis, vastus intermedius and vastus lateralis, or quadratus femoris a.k.a. "quads," and also your sartorius.
Note also that some of the muscles that work at the knee also work at the hip, not the least of which are your quads and hamstrings.
What exercises make these muscles work?
Same exercises as for movement at the hip, for the most part. Exercises for rectus femoris also work the rest of the quad muscles, exercises for biceps femoris also work the popliteus.