I try to get into optimal transport by reading Villani's book "Optimal Transport - old and new" and I came across this definition. Unfortunately, I dont really have an intuition on what motivates them (lines (5.6)-(5.8)). They seem pretty mysterious to me.
The motivation is as follows. For the quadratic cost, i.e. $d(x,y)^2$, there are a bunch of basic optimal transport facts which can be proved using techniques/objects from convex analysis, including: Legendre transform, subdifferential, etc. It would be nice to be able to run those proofs for costs other than $d(x,y)^2$; the solution turns out to be to define a generalization of convexity relative to a given cost $c(x,y)$, such that, in the terminology of $c$-convexity, $d(x,y)^2$-convexity is just regular convexity.
In particular, the $c$-transform is the "right" generalization of the Legendre transform, the $c$-subdifferential is the "right" generalization of the subdifferential, etc.
I can't remember if this is emphasized in Villani's book, but definitely in Santambrogio's book the special, convenient proofs that use convex analysis techniques in the quadratic case are presented, so you can try and compare how the generalization goes through when one switches to a general cost.