If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid.
The circulation is defined as the integral of the velocity around a closed curve. An intuitive way of thinking about this is that if the fluid flows in the same direction everywhere, then integrating the velocity around any closed curve will give you 0. Indeed, you will integrate in the direction of the flow on some interval of the curve, but you will also have to "come back" and integrate against the flow, so the contributions will cancel. If on the other hand the fluid rotates in a vortex, then as you integrate the velocity around a circle centred on the axis of rotation, you pick up a positive contribution to the integral at each point.
Further from wikipedia:
Suppose the vector field describes the velocity field of a fluid flow (maybe a large tank of water or gas) and a small ball is located within the fluid or gas (the centre of the ball being fixed at a certain point). If the ball has a rough surface it will be made to rotate by the fluid flowing past it. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the value of the curl at this point.