Dirac delta distributions only make sense in an integral. For example, one could want the total work done by the force:
In this case, that would mean that all the force is acting on a single point. An integral over a single point gives $0$ ($\int_1^1 = 0$). But since we do want the force to act on the object and do work, we don't want this. In order to have a finite integral value one need to "infinitely concentrate" the force on that point. Thus, the delta dirac distribution.