What is my cousin's goal in trying to speak with me privately, and what should I do about it?

If your male cousin (son of your paternal aunt) is a stranger to you (i.e., not a mahram), it is not permissible to respond to his request to speak with you in private, because khalwa (private seclusion) between two non-mahrams is forbidden, as the Prophet (peace and blessings be upon him) said: "No man should be in private seclusion with a woman unless there is a mahram with her." If his intention is marriage, then he should propose to you through your guardian.