I see that there are a few ways to prove this, but I don't know why my method isn't working.
I set ${^{n - 1}C_r} = \dfrac{(n - 1)\cdot(n-2)\cdot\cdots\cdot(n - 1 - r + 1)}{r!}$ so the last term in the numerator is $n-r$. Call the numerator $N_1$.
Then ${^{n - 1}C_{r - 1}} = \dfrac{(n - 1)\cdot(n - 2)\cdot\cdots\cdot(n - 1 - (r - 1) + 1)}{(r - 1)!}$ so the last term in the numerator is $n-r+1$. Call this numerator $N_2$.
Then the sum is $\dfrac{N_1 + r\cdot N_2}{r!}$.
I don't know if I made a mistake or I'm failing to see how this equals $^nC_r$.