The easiest way to think about it is to realize that the phosphates on the sides are strongly negatively charged. If DNA stayed in a straight "ladder" form, there would be a whole bunch of negative charges in the same area (think of a cloud of negative charge up/down the sides of the ladder), with no negative charges in the area of space between the sides of the ladder. Negative charges repel each other, which drives the phosphates to try to get away from each other. The bonds of the sugar phosphate are too strong to stretch to get away up or down, but they can shift to either side which is less negative. This causes the entire thing to twist up into the helix shape. Because there's two strands, it's a double helix.