I will describe a geometric interpretation of bordered Heegaard Floer invariants in the case of a manifold M with torus boundary. In particular these invariants, originally defined as homotopy classes of modules over a particular algebra, can be described as collections of decorated immersed curves in the boundary of M. Pairing two bordered Floer invariants corresponds to taking the Floer homology of immersed curves; in most cases this simply counts the minimal intersection number. This framework leads to elegant proofs of several interesting results about closed 3-manifolds. As one example, I will prove a lower bound for the complexity of Heegaard Floer homology of a manifold containing an incompressible torus, reproving and strengthening a recent result of Eftekhary.