Step 1: Recognise the clinical picture.
Repeated gripping and hammering, pain on the outer (lateral) side of the elbow, and pain on wringing a sponge (which needs a firm grip with the wrist held extended) are the classic description of lateral epicondylitis, commonly called tennis elbow.
Step 2: Recall the anatomy of the lateral epicondyle.
The common extensor origin on the lateral epicondyle of the humerus gives rise to the wrist and finger extensors. The muscle most consistently involved in tennis elbow is extensor carpi radialis brevis, with extensor carpi radialis longus also commonly affected, since these muscles are heavily loaded whenever the wrist is stabilised in extension during a strong grip, such as swinging a hammer or wringing out a sponge. Repeated microtrauma at their tendinous origin causes degeneration and pain at the lateral epicondyle.
Step 3: Rule out the other options.
Biceps brachii and supinator act mainly at the elbow and forearm for flexion and supination, and their pathology, such as biceps tendinopathy, does not cause lateral epicondyle pain. Flexor digitorum superficialis arises from the medial epicondyle, so its overuse causes medial elbow pain (golfer's elbow), not lateral pain. Triceps brachii and anconeus act on elbow extension from the posterior surface, and their overuse gives posterior elbow pain, not the lateral pain described here.
Step 4: Final conclusion.
The muscles most likely involved are extensor carpi radialis longus and brevis.
\[ \boxed{\text{Extensor carpi radialis longus and brevis}} \]