The identity just splits the ${n+1 \choose k}$ subsets into two types: those subsets that contain a given element, and those subsets that do not contain this given element.
Let's call the set of $n+1$ elements $S$ and pick one element $x \in S$.
So first, how many subsets of $S$ of size $k$ contain $x$? Well apart from $x$ there are $n$ elements in $S$. Formally, $|S-\{x\}| = n$. We want subsets of size $k$ and we already have $x$ in our subset. Thus we need to pick $k-1$ elements from $S-\{x\}$. Thus the total number of these subsets is ${n \choose {k-1}}$.
Second, how many subsets of $S$ of size $k$ do not contain $x$?
Now we need to pick $k$ elements from $S-\{x\}$, which gives ${n \choose k}$ subsets.
All subsets of $S$ of size $k$ either contain $x$ or not, so we have counted all of them, and found that
$${n+1 \choose k} = {n \choose k} + {n \choose k-1}.$$