If $y_1,\ldots y_k$, where all $y_i \ge 0$, is a solution of the equation
$x_1 + x_2 + \ldots x_k = n, x_i \ge 0,i=1 \ldots k$
then $y_1 + 1, y_2 + 1, \ldots, y_n + 1$ is a solution to the equation
$x_1 + x_2 + \ldots x_k = n+k,x_i \ge 1,i=1 \ldots k$
of which there are $\binom{n+k-1}{k-1}$ by your own formula.
Reversely, for any solution for the second equation, substracting 1 from each (which is possible) gives a solution for the first one, so the operation of adding 1 to each solution is a bijection between the solution sets of these 2 equations.