Donaldson-Thomas invariants in mathematics are a virtual count of sheaves (or possibly objects in the derived category of sheaves) on a Calabi-Yau threefold. In physics, sheaves (and more generally objects in the derived category) are considered as models for D-branes in the topological B-model and Donaldson-Thomas invariants are counts of the BPS states of various D-branes systems. For example, the "classical" DT invariants that are considered by MNOP count ideal sheaves of subschemes supported on curves and points. You will hear physicists refer to such invariants as "counting the states of a system with D0 and D2 branes bound to a single D6 brane". The single D6 brane here is the structure sheaf $\mathcal{O}_X$ and the D0 and D2 branes form the structure sheaf $\mathcal{O}_C$ of the subscheme $C$ (which is supported on curves and points) and the term "bound to" refers to the map $\mathcal{O}_X \to \mathcal{O}_C$ because they are replacing the ideal sheaf with the above two-term complex (which are equivalent in the derived category. Note that the $k$ in D$k$-brane refers to the (real) dimension of the support.

There is a discussion of the meaning of the motivic DT invariants in physics in the paper "Refined, Motivic, and Quantum" by Dimofte and Gukov (http://arxiv.org/pdf/0904.1420) where the basic claim is that the motivic invariants and the "refined" BPS state counts are the same. "Refined" here refers to the way you count BPS states. BPS states are certain kinds of representations of the super-Poincare algebra and "counting" means just finding the dimension of these representations (I think that little book on super-symmetry by Dan Freed has a good mathematical discussion of this). Sitting inside the super-Poincare algebra is a copy of $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$ and normally one looks at the action of the diagonal $\mathfrak{sl}_2$ on the space of BPS representations and finds the dimensions of the irreducibles, for the "refined" count, you look at both copies of $\mathfrak{sl}_2$. The generating function for the dimensions of these representations thus gets an extra variable which is suppose to correspond to the Lefschetz motive $\mathbb{L}$ in the motivic invariants.

As for the DT/GW correspondence, I'm afraid that I don't really understand the physicist's explanations. There is a few paragraphs in MNOP (presumably written by Nekrasov) about it and I think that physicists regard it as well understood, but I haven't found something that I can understand. Let me know if you do.