The spin keeps the bullet stable.
Once you get something moving in a particular direction, it continues moving in that direction until something stops it. Newton's first law of motion and all that.
Motion is velocity and velocity implies momentum (momentum is mass times velocity), so if the bullet moves sideways, it must have momentum in that direction. If it has momentum in a given direction, Newton's first law says it will keep moving that direction until something stops it.
If you roll a ball across the floor and blow on it to change its direction, it doesn't immediately resume its original direction once you stop blowing on it--it will keep rolling on the new path until something else causes it to change direction again.
So once a bullet is (moved sideways by/gains sideways momentum from) a crosswind, even if the crosswind goes away, it will still keep moving sideways until air friction stops it. Since it's moving sideways relatively slowly, there's very little air resistance in that direction to speak of and it's not going to be in the air long enough to be slowed significantly in terms of crossrange velocity.
When looking at position plots (like the one under discussion), straight lines mean velocity with no acceleration. Curved lines mean acceleration is being imparted.
So, on the plot, the parts of the traces that are curved are the areas where the bullet is being accelerated sideways by wind. The wind is applying a force to the bullet (or you could say accelerating it, since acceleration = force/mass) which is increasing its velocity in the direction of the wind. The straight lines (whether slanted or horizontal) show the areas where there is no acceleration (force) on the bullet due to wind. A horizontal line means no sideways velocity, a slanted straight line means a sideways velocity.
It may seem a bit strange to think about sideways velocity being measured in inches per yard, but if you think about it, the horizontal axis could just as easily be labeled in terms of time (time of flight) and then you'd have inches per second which is obviously a measure of velocity.