Looks to me it's a formula for calculating the % change in cross-sectional area for a % stretch lengthwise. Cross-section area is a function of the width (or perhaps width x length). But width is compressed as length is stretcehd and that has to be taken into account while calculating the change in area.
Having written all this, I don't get why a squared term is involved, because while the calculation of an area may involve a square, the change in such an area would be linear.
From a Yahoo search on "Poisson's ratio":