In a rollercoaster, for a passenger to remain in contact with his seat, the centripetal force must equal the weight. Why is this?

Yes, well actually centripetal force must be less than the person's weight for him to remain in his seat. I think it's better to think about this in terms of acceleration: at the top of an arc of the roller coaster the centripetal acceleration downward of the coaster must be less than gravitational acceleration in order to stay in the seat.