Total no. of functions is $n^m$. [For every value from ${1,2....,n}$ there are m possibilities].
We choose an m-element subset of n, number of such subsets $=\binom{n}{m}$
Since it is an increasing function, we can order the elements in only one way, i.e in ascending order.
Therefore , probability is $\frac{\binom{n}{m}}{n^m}$