Choice (A) has an ungrammatical construction, because the main clause has two grammatical subjects, "immigrants" and "they." The pronoun "they" is grammatically redundant. We can eliminate choice (A).
In choice (B), the pronoun "they" doesn't have a sensible reference. Its only possible reference is "immigrants," but the "knowing" described in the sentence isn't done by the immigrants--it's done in general: the immigrants are known as the Pilgrims. Therefore, we can eliminate choice (B), and also choice (C), on the same grounds.
In choice (D), the infinitive "to be" incorrectly indicates that the immigrants will be known as the Pilgrims in the future, or that they should be known as the Pilgrims. Either way, the unnecessary infinitive distorts the meaning of the sentence, so we eliminate choice (D).
Choice (E) is simple and accurate. The phrase "known as the Pilgrims" modifies "immigrants," creating a logical reference. No additional words are required.
The correct answer is (E).