B is a prepositional phrase as it contains a prepositon as the head (throughout) followed by an NP (nominal phrase) which functions as the object.
A, C and D are not phrases but sentences. Consequently, they are composed of two parts: a subject and a predicate. For example, in "but they never stopped", the personal pronoun "they" functions as the subject of the sentence, that is, it's what the sentence is about. The rest of the sentence is the predicate, it tells something about the subject.
A is grammatically incorrect as it stands for an incomplete sentence. It contains the subject (they) and a main verb functioning as the head of the verbal phrase but it does not have the subordinate clause which should followed after the verb for the sentence to be considered correct. On the contrary, D is gramatically correct for it is a complete sentence. However, it is not a prepositional phrase because it is not a phrase but a sentence. The pronoun "it" is the head of the phrase and it is not a preposition. The verbal phrase "was a happy time" stands for the predicate, making the sentence grammatically correct. Said VP (verbal phrase) takes a nominal phrase as the object ("a happy time").
To sum up, prepositional phrases are made up of a preposition functioning as the head and its object. It can also contain modifiers. They take a nominal phrase as the object. That is why B is the correct answer. "Throughout" is the head of the PP (prepositional phrase) taking the nominal phrase "his life" as the object.
