What is the geometric interpretation of the rowspace?

As Ted Shifrin states in his comment to your question, the rowspace of a real matrix is the orthogonal complement of its nullspace. In fact, we have the following relationships for any (not necessarily square) real matrix $A$: $$\mathscr{C}(A^T)=\mathscr{N}(A)^\perp \\ \mathscr{N}(A^T)=\mathscr{C}(A)^\perp.$$ There is an equivalent pair of statements about a linear map, its adjoint map and annihilators of subspaces.

One consequence of these relationships is that the rowspace of a matrix is a maximal-dimension subspace of the domain for which the linear map defined by the matrix is injective. Another possible way to view the rowspace is as the space of representative elements of the quotient space $\mathbb R^m/\mathscr{N}(A)$, where $m$ is the number of columns of $A$. In light of this, the rowspace of a matrix can be considered the “natural” preimage of its column space.