DNA twists primarily because of van der Waals forces and the hydrophobic effect.
Van der Waals forces draw the bases closer to each other and the DNA twists to accommodate their positions. Think of a ladder made out of tooth picks and marshmallows. If the rungs of the ladder represent the bases, the only way to get them closer to each other without disconnecting the marshmallow joints is to twist it.
The hydrophobic effect originates from waters affinity for itself and dislike for non-polar substances. This causes the non-polar substances to be excluded as much as possible from interactions with water. That being so, it is more favorable for the generally non-polar bases to be within closer proximity of one another.