What do the dimensions of circulation mean, and how is circulation related to action?

The best way to gain intuition about circulation is to think of it as a measure of the number of vortex lines through a surface. By Stokes' theorem, $\int_C \vec{v} \cdot \vec{dr}$ = $\int_A (\vec{\nabla} \times \vec{v}) \cdot \hat{n} \, dA$. The quantity $\nabla \times \vec{v}$ called the vorticity, and you are threading the surface $A$ with it. From this perspective, the units of circulation make sense as [vorticity]$\times$[area].

Note the analogy you can make to Ampere's law in E&M. Vorticity is analogous to the electric current density, and velocity is analogous to the magnetic field.

Circulation is a useful concept in fluid dynamics primarily because it obeys a conservation law in the absence of viscosity (Kelvin's circulation thoerem). This conservation law is stated without reference to the fluid's density, and for this reason I don't think there is much physical significance to the quantity $\int_C \rho \, \vec{v} \cdot \vec{dr}$ or its units of action/volume.

EDIT: I thought about this some more, and now I realize that $\int_C \rho \, \vec{v} \cdot \vec{dr}$ is physically meaningful after all - it is the angular momentum density per volume. So instead of understanding the units via action as you had suggested, think about angular momentum instead. And the conservation law for circulation that I mentioned above (Kelvin's) can be interpreted as conservation of angular momentum for the fluid.

Don't confuse circulation with the vortex that produces it. Circulation is irrotational flow, vortices are rotational and are found in the boundary layer. The flux of the vorticity is circulation. In aerodynamic applications, I find it helps to understand its physical meaning by dividing the circulation by the distance it acts over (i.e. the chord of the wing) to get the imparted velocity to the fluid. This velocity multiplied by the airstream velocity and by air density gives you the pressure difference that lifts the wing.