In many places around the world diners consider roasted insects a delicacy what is the function of the underlined phrase in the sentence

You forgot to mention which is the underline phrase in the sentence. However I dug up on other sources and I believe the correct underline phrase is:

“In many places around the world”

Now the function of this underlined phrase is called as “adjective prepositional phrase”. A preposition is something which connects a word in a sentence to the other words. In this case, the underlined phrase connects diners to the rest of the words to give meaning to the sentence. See, if we remove the underlined phrase, it would become “Diners consider roasted insects a delicacy” which does not really make sense since we do not know which diner the sentence is referring to. Since the underlined phrase also describes the word diner, hence it is also an adjective and therefore an “adjective prepositional phrase”.

adjective prepositional phrase