The numerator of the left side is $n(n-1)\cdots (n-k+1)$.
The numerator of the right side is $n(n-1) \cdots (n-k+1)(n-k)$.
Dividing both sides by $n(n-1)\cdots (n-k+1)$, the left's numerator becomes $1$, while the right's numerator becomes $(n-k)$.
Similarly, the denominator on the left side is $k(k-1)\cdots 1$.
The denominator on the right side is $(k+1)k(k-1)\cdots 1$.
Multiplying both sides by $k(k-1)\cdots 1$, the left's denominator becomes $1$, while the right's denominator becomes $k+1$.
This results in: $1/1 = (n-k)/(k+1)$.
Does this clear things up?