Solution:

We claim that the sum is equal to $m^{n}$.

We note that $m^{n}$ counts the number of ways to color $n$ objects each with one of $m$ different colors, so it suffices to show that the left side counts the same thing.

We can consider cases based on how many different colors get used. If $k$ colors are used, then there are $f(n, k)$ ways to choose how to split the objects into subsets based on color, then $m$ ways to choose the color of the first subset, $m-1$ ways to color the second one, and so on. Summing over $k$ from $1$ to $n$ gives the desired sum.