How was Jefferson's development of "the grid" involved in the expansion/strengthening of the U.S as a nation?
When you refer to "the grid" I assume that you are talking about Jefferson's idea of land in the "Northwest Territories" up into the townships and sections, half sections, etcetera.
If so, I think that there are at least a couple of ways in which this helped strengthen and expand the US.
First, I think that this helped by bringing order to these new areas. It allowed there to be a very well-defined survey in the new areas so that there would not be fighting over what land belonged to whom. This allowed settlers to move in and be confident that they truly owned their claims. (I should also mention that the low prices for the land were important in allowing settlement.)
Second, I think that Jefferson's idea of dedicating the income from one section per township to education was important. It helped to emphasize the idea that public education was important, even in the frontier areas.