How to interpret couplings in optimal transport?

Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable mappings }T: X \rightarrow Y \text{ and } T_\#\mu = \nu$$has a relatively straightforward interpretation: We try to find a measurable map $T$ that minimizes the cost to move mass from $x$ to $T(x)$. Now, the Kantorovich problem to$$ \text{minimize} \quad \int c(x,y)\pi(dx,dy) \quad \text{over couplings } \pi \text{ with first and second marginals } \mu \text{ and } \nu \text{, respectively,}$$I find to be much harder to interpret as a real `mass transfer' problem; If $\pi^\star$ is an optimal coupling to the Kantorovich problem, what does $\pi^\star$ tell me where how much mass really goes? How do I interpret the Kantorovich problem?