“Out of respect for the rest of the class”
The full sentence is: "Out of respect for the rest of the class, please arrive on time."
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In traditional terminology, a clause must have a subject and a finite verb:
If you value courtesy and respect for your classmates, you will arrive on time.
This sentence has two clauses:
A phrase is a group of words performing the same function in a sentence:
Your example sentence
Out of respect for the rest of the class, please arrive on time.
begins with three prepositional phrases. There is no verb. The prepositions are in bold:
Thus it can only be termed an introductory phrase.