What does 'laterally aligned' mean in this description?
The only meaning that I can conceive of is that "laterally aligned with the centerline" is a fancy way of saying "on the centerline"—or, at least, keeping the distance from the centerline under control, so that you don't drift off of the runway.

The paragraph seems to be describing the following technique for lateral control during a crosswind takeoff:


Use the rudder to keep the heading (the direction the nose is pointing) parallel to the runway centerline.
Use the ailerons to control the bank angle so that the airplane stays approximately over the runway centerline.


The reason the Airplane Flying Handbook uses the word "laterally" here is that the ailerons are being used to control the airplane's lateral (left-to-right) motion.

Some comments speculate that the paragraph is really saying that the ailerons should be used to keep the wings level. That doesn't make sense to me; in a crosswind, if you're using rudder to keep the nose parallel to the centerline, and you're keeping the wings level, then you'll drift downwind off the runway.