Like many of his contemporaries did, Bob Dylan wrote songs that became anthems for a generation of antiwar activists.
A is incorrect from a grammatical stand point as we use Like only to compare nouns. Like is used here to compare the actions of Bob Dylan's contemporaries to Bob Dylan
(A) Like many of his contemporaries
did, Bob Dylan
wrote songs
B is incorrect for similar reasons, but sentence also illogically communicates that Bob Dylan wrote songs that were similar to his contemporaries. Songs that are similar to humans? hmm?
(B) Bob Dylan wrote songs like many of his
contemporariesC is correct- AS is used to compare the actions here.
(C) As did many of his contemporaries, Bob Dylan wrote songs
D illogically compares the songs of Bob Dylan's contemporaries to Bob Dylan himself
(D) Like the songs of his contemporaries, Bob Dylan wrote songs
E illogically communicates that Bob Dylan's contemporaries did something. The "he" cannot refer correctly to the possessive here as it is a subject.
Only possessive pronouns can refer back to possessive nouns.
(E) As did many of Bob Dylan’s contemporaries, he wrote songs