The multinomial theorem states that:
$$(x_1 + x_2 + \dots + x_m)^n = \sum_{k_1 + k_2 + \dots + k_m = n}{n \choose {k_1, k_2, \dots, k_m}}\prod_{1 \le t \le m} {x_t}^{k_t}$$
The number of monomials is the number of solutions to this equation for each $k_i \ge 0$:
$$k_1 + k_2 + \dots + k_m = n$$
Is there a straight forward way to count the number of solutions to the above equation?
As you noted, the number of monomials in the expansion of $$(x_1 + x_2 + \cdots + x_m)^n$$ is the number of solutions in the nonnegative integers of the equation $$k_1 + k_2 + \cdots + k_m = n$$ where $k_j$ denotes the exponent of $x_j$, $1 \leq j \leq m$. The number of such solutions is the number of ways $m - 1$ addition signs can be placed in a row of $n$ ones, which is $$\binom{n + m - 1}{m - 1}$$ since $m - 1$ of the $n + m - 1$ symbols are addition signs.