It's better to work in 1-forms to see what's going on. We can think of a one-form $\lambda$ as a section $s:X \to T^*X$ by pulling back the Liouville form $s^*\theta = \lambda$. The symplectic form is $d\theta$ so the "curl" from this roundabout construction is $s^*d\theta = ds^*\theta = d\lambda$, so curl is just dual (in the sense that 1-forms correspond to vector fields by line integrals) to the exterior derivative, which is ubiquitous in geometry and physics.
Here is a geometric interpretation of the curl: it's a 2-form, so it takes a value on each tangent plane. If one projects the vector field onto this plane and computes the usual 2d curl it will be the value of the 2-form curl on this tangent plane.