The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R) such that the group contains both T and T.
The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups, which only repeat along a single axis.
(It is possible to generalise this situation. One could for example study discrete groups of isometries of R with m linearly independent translations, where m is any integer in the range 0 ≤ m ≤ n.)