Answer: This is an example of an adverb phrase.
Explanation: An adverb phrase is a phrase which acts like an adverb and can express time, manner, reason, location, etc. In this specific example, the adverb phrase with enthousiasm expresses manner as it answers to the question "how?".
Another example of an adverb phrase is the following:
They looked at me with suspicion.
With suspicion is an adverb phrase which expresses manner.