Answer: Prepositional Phrase.
If it were a verbal phrase you would see a verb turned into a phrase like the verb "glare" to "glaring" you might see "The man with dark glasses glared at me glaring under them."
And independent clause is just a sentence. I must be able to stand alone. In this case "from under them" cannot stand alone as it lacks a subject or a verb.
And appositive phrase is where the nouns are used to describe other nouns. A noun is a person, place, thing or idea. Example "The man with the dark glasses glared at me, a foreigner."
A prepositional phrase is a phrase that begins with a preposition. Examples of prepositions are words like "over, under, to, from, up, down"
In this case prepositional phrase is your winner!