To control the position of the paper on a printing press with a high degree of accuracy several guides are used. These are called lays. As a sheet is finally aligned before being printed it is gently pressed (by a moveable lay) against a semi-fixed side lay and also an end lay. These determine that every sheet shall be positioned ~ and subsequently printed ~ in exactly the same way.
If all the sheets are off-centre or crooked then they can all be trimmed in a guillotine ~ by setting the lay edge against the guillotine stops. The print on every sheet will bein the same place ~ called being in register. Laying off ensures that every sheet has the print at exactly the same distance from each of the lay edges ~ regardless of the size of the paper.
The term takes the same meaning in other situations. When bookbinding sheets are knocked down on a hard surface it sets them all evenly on one edge ~ the lay edge. Subsequent measurements ~ for punching holes ~ are then always taken from the lay edge. If some sheets are of different size then that will not matter.