In the sentence, "Tell me frankly why you did this," the underlined part, "why you did this," is an adverbial clause because it functions like an adverb, explaining the manner or reason of the action described in the main clause, "Tell me frankly". 
Here's why:
It modifies the verb "Tell"
: The clause "why you did this" provides the reason for the telling, similar to how a single adverb like "why" would provide the reason or manner. 
It answers a question
: You can think of this clause as answering the question, "Tell me frankly for what reason?". 
Therefore, it acts as an adverbial clause, describing the reason or purpose for the action in the main sentence.