Of the mass $\mu(A)$ in $A$ a fraction $\pi(A \times B)$ is transported to $B$, so you can think of this as a randomized transport map. A basic example to think of is $\mu=\delta_0$ and $\nu=(\delta_1+\delta_{-1})/2$. Half the mass at 0 is sent to 1 and half is sent to -1. You can get a better intuition from reading more about construction of couplings:
[1] Lindvall, Torgny. Lectures on the coupling method. Courier Corporation, 2002.
[2] Thorisson, Hermann. "Coupling methods in probability theory." Scandinavian journal of statistics (1995): 159-182.