When doing a chin up (in the phase where the individual is moving up toward the bar), the individual's elbows and wrists flex--assuming a reverse grip on the bar. Flexion occurs when the angle between the bones of a joint decreases. If you bend your arm at your elbow, you are flexing your elbow. If you bend your wrist so that your palm gets closer to your forearm, you are flexing your wrist.
In order to perform these particular muscle actions, several muscles must be employed: prime movers (a.k.a. agonists), which are responsible for the majority of the work associated with the action; antagonists, which perform the actions opposite to the agonists by relaxing when the agonists contract; synergists, which aid the agonists with the action being performed; and fixators, which stabilize nearby joints preventing movement in them.
Your question specifically asks about the prime movers or agonists. The main muscle involved in flexion of the elbow is the biceps brachii muscle. The brachialis and brachioradialis would be synergists. The flexor carpi radialis, flexor carpi ulnaris, and palmaris longus are prime movers when flexing the wrist. Flexing of the wrist is more subtle than the flexing of the elbow.
The radioulnar joint has a fixed position during this exercise and thus stabilizing muscles are active.