This statement conveys the idea that being with someone special brings out the best in the speaker. "I Remember Who I Am When I'm With You" signifies that the presence of the person addressed helps the speaker to rediscover and embrace their true self.
It implies that this person's influence is not just on the speaker's emotions but also on their sense of identity and self-awareness. In essence, this quote beautifully captures the idea of a deeply enriching and transformative relationship where the person addressed has a profound impact on the speaker's self-perception and self-confidence.