You are correct; Ernie is not asking a question, but making a statement.

As you may know, most declarative sentences in English follow subject-verb-object (SVO) order, but they can be reversed or altered. Inversion can be used for emphasis as well as in questions, conditionals, comparisons, and so on. As a rhetorical device it is known as anastrophe.

Ernie could say I am thirsty, but as the clip shows, he seems to want to make it well-known that he is thirsty, or that he is very thirsty, so he chooses the more emphatic form— and furthermore introduces the sentence with boy, in Merriam-Webster's words, used to express intensity of feeling. The humor lies in his continuing to bellow about thirst afterwards, when not only is emphasis unnecessary, but the very pronouncement of his former thirst is itself unnecessary.

This kind of inversion is not altogether uncommon; consider examples like the following:

Are you excited to visit your grandmother?
Am I ever!

On no point did we agree.

The English language I understand; English speakers are another matter.

"Boy, am I thirsty."

Boy here is a simple exclamation, emphasizing the sentiment that follows. He could have said, "Oh" or "My" or "Gosh" or any number of other mild exclamations in its place.

"Am I thirsty."

This type of phrase is used to assert that one is thirsty. He could have said, "I am thirsty," and it would have meant the exact same thing.

The effect of it is to use the phrasing of a question, but one with an answer that is so obvious that it is an assertion.

Here is passage in a book, "The Joy of Grammar" which refers to this very type of construction. The author calls them "speech act constructions" but I have not heard them given a formal title elsewhere. It's not an uncommon thing for someone to use in AmE speech, particularly with familiar people.