In your first example, the original adjective was 暖かい, which is in i-adjective. The adjective was turned into an adverb by removing the い and changing it with a く, so it modifies the なりました. Other simple example is 良い, adverb form 良く, example sentence 良くなりました (Lit. : It became good) Your 2nd example was using きれい. It is actually a na-adjective, eventhough it is ended with い. We will say for example きれいな人 (A beautiful person). The adverb form for na-adjective is different. Mostly it will remove the な and adds particle に Like in your example 手がきれいになりました