The verb 'entered' requires a noun or a pronoun (in subjective case) as its subject.
'Me' is a pronoun in the objective case, which is incorrectly used in this sentence in place of the subjective pronoun 'I'.
So, 'me' should be replaced with 'I' to rectify the error.
The correct sentence is: Everybody stared when I entered the drawing room.
Hence, option (a) is the answer.
Note: Subjective Pronouns: The pronouns I, you, we, he, she, it, and they are known as subjective pronouns because they act as the subjects of verbs: We went to the party.
Objective Pronouns: The pronouns me, you, us, him, her, it, and them are called objective pronouns because they act as the objects of verbs and prepositions: Anjali drove us home.