What is the physical meaning of circulation found in Kutta condition?
Circulation of a fluid around an object by itself will produce no lift. The classic example of this is the spinning cylinder with no other airflow. Viscosity will cause the fluid near a cylinder rotating clockwise to circulate in a clockwise direction around the cylinder. If a left to right horizontal flow is introduced there will be a vector sum of the two flows. This results in the stagnation points near 8 o'clock and 4 o'clock (as opposed to the cylinder with no rotation in the left to right flow having the stagnation points at 9 and 3 o'clock.) The net result of this is the Magnus effect where lift is generated in the 12 o'clock direction.  

In your first diagram (typical inviscid flow) there is no circulation. The shape of the airfoil in viscid flow causes the trailing stagnation point to move to the trailing edge (second image -- the Kutta condition). This has the same effect on the airflow as the spinning of the cylinder, in that it creates a clockwise circulation about the airfoil.  

The line integral describes, for an arbitrary closed contour around the object, the dot product of the fluid flow velocity vector with the vector path moving around the contour. The simplest contour to analyze is created by following flow streamlines above and below the airfoil and connecting them before and after the airfoil with lines perpendicular to the streamlines.  

Since the dot product of perpendicular vectors is 0, the integral along perpendicular portions of the contour are 0. The dot product of parallel vectors  is just the multiplication of the scalar values, and since the direction of the contour is reversed between the upper and lower streamline the effect is adding one and subtracting the other. Due to differences in lengths and the different flow speeds (Bernoulli...) along the contour, the integral is non-zero. This number represents the net effective circulation about the airfoil (total flow minus the horizontal flow.)

The interesting thing is that if you extend the contour behind the airfoil far enough to enclose the wake of the airfoil from the beginning of movement, the circulation will be zero as the circulation of total wake is the vector opposite of the circulation about the airfoil.