Why does ankle popping occur after a sprain healed?

Depends. Ankle 'popping' can occur after sprains due to scar tissue, cartilage damage, or possibly from a dislocating tendon. No treatment is needed if the popping causes no pain. If pain is associated, i recommend you see an ankle specialist for evaluation.
Possible. Ankle popping is not a diagnostic sign in and of itself. Some people naturally pop their joints easily, others don't. If you continue to have pain or limitation in the range of motion along with popping, it warrants checking to make sure there is no other abnormality, such as missed hairline fracture. If in doubt, go to your primary care provider or the one who took care of your sprain.