What is the derivation for the expression of depression in freezing point?
To derive it, you have to know how an equilibrium depends on concentration and on temperature.
At the freezing point, pure liquid and the solid are at equilibrium. If you lower to concentration of the liquid by adding solute, you disturb the equilibrium. If you lower the temperature by the right amount, you are back at equilibrium.
The relationship you get is:
$$x_{A}=\frac{\Delta H_{m}}{RT^{\text{2}}}\ \Delta T$$
This is from a General Chemistry text that also does not derive it. With some algebra, you can express $K_f$ in terms of the enthalpy of fusion comparing this and the equation given by the OP.