Assume a fit of a straight line through 10 points, each point with an error. Then we say that the line has 2 free parameters and the number of degrees of freedom is 10 – 2 = 8.
For a kinematic fit we would say that there are 10 measured variables (the 10 y values), 10 fitted variables (the 10 y values along the line at the same 10 x positions), and 2 unknown variables (the slope and y-intercept of the line). For each fitted variable there is a constraint: that the fitted value lies along the line, i. e., that y_fit,i = ax_i + b. So there are 10 constraints. We say that we have an 8-C fit since the number of constraints minus the number of unknown variables is 10 – 2 = 8.
So the C-ness of the kinematic fit corresponds to the number of degrees of freedom in the chi-squared fit.