Your two conditions of "for an electrical engineering student" and "have proofs but in an UN-intimidating approach" are a little contradictory. That's not your fault, but the fault of many electrical engineering students, who care nothing for proofs. My standard recommendation is Calculus: An Intuitive and Physical Approach, by Morris Kline. He has great explanations, and great applications. He is a true teacher. It is not the most rigorous book on the planet, but that wasn't his goal in writing it.
A bit more proof-heavy would be my favorite current standard book, which is Stewart's Calculus: Early Transcendentals. It's a solid text, and has everything you need.