Why does it ache terribly on the back of the knee when stretching, standing when my legs are straight and I do tilts with my torso, or sitting just as I reach for my socks?

Alexander, hello! In addition to stretching the tense muscles of the back of the thigh, it will not hurt to additionally train the antagonist muscles (opposing) the hamstrings (the largest and most powerful muscle of the back of the thigh). The main antagonist will be the quadriceps muscle of the thigh (the largest and most powerful muscle of the front of the thigh). Additionally, I would recommend strengthening the abdominal muscles to compensate for the forward tilt of the pelvis that is likely to be present. This is important in order to further avoid problems with the lower back , which are a frequent companion in the physiological shortening of the muscles of the back of the thigh.

I can only add that regular stretching will help you to remove this tension. Not soon, not quickly, but they will help. The main thing is not to force events and not bend over in jerks, as in physical education, it would not be better - there is a risk of injury. Slow bends like yoga will help a lot.

In this place, the muscles of the posterior group of the thigh pass, which are attached to the pelvis at one end and to the bones of the lower leg at the other. Accordingly, when bending forward, the pelvis, rotating, pulls the muscles towards itself, and the greatest tension occurs in the tendon zone behind the knee. That's why it hurts.