Some of our mathematic operations does not only depend on calculations of their value at a fixed set of known points.
For example we have optimization, integration and integration of differential equations.
For optimization we want to search for the best parameters, which gives the smalles function-value. This is most conveniently done if we have an optimization-function that can handle a large range of input-functions - that is where the function-handles comes in, an optimization-function that can handle a function-handle can optimize any function, otherwise we'd have to produce tailormade optimization-functions for each optimization-problem (this tailor-making might not be terribly complicated, perhaps it would be possible to just change some function-calls and input-arguments at a couple of places in this optimization-function, but many run multiple optimizations per day and it is way mode convenient to use function-handles).
For integration we have the same situation, it is way easier if we want to integrate some complicated function to be able to call the integration-function with a function-handle (for multi-dimensional integrals we might even want to have some of the integration-boundaries be curves that are easiest to handle with function-handles), for example:
Q1 = integral(@(x) x.^2,0,3)
Q2 = integral(@(x) exp(-x^2),0,inf);
For differential equations we have the same situation, the ODE and PDE-integrating functions use very clever schemes that evaluate the differential equation and its solution at many intermediate positions, the best way is for such functions to accept a function-handle (or multiple function-handles for the PDE with initial and boundary-conditions) that they can evaluate as is required.
Then it is possible to imagine additional use-cases as well, I remember using function-handles for some other problems but cannot remember exactly what...
...however, once you've grasped the concept you should also remember that it is not the only programming-pattern you should use, some other problems are better solved with other tricks...
Hopefully this clarifies the benefit of function-handles.