Let's look at the first sentence:
Tell me what the exchange was like.
The nominal clause ("what the exchange was like") functions as the direct object of the verb "tell". It contains a subject ("the exchange"), a simple predicate ("was"), and a prepositional phrase ("like what", with the preposition postponed, which is often allowed in subordinate clauses). Everything makes sense, and this sentence is grammatical.
Now let's look at the second sentence:
Tell me how the exchange was like.
The preposition "like" no longer has an object (complement)! "How" is either an adverb or conjunction (depending on your interpretation), but neither of those parts of speech typically functions as an object. Because prepositions normally require objects, this sentence is incorrect. We could fix it simply by omitting the preposition:
Tell me how the exchange was.
This is now grammatical, but for the meaning that you intend, the original sentence ("tell me what the exchange was like") is probably better, as KB has noted in a comment.