In the left sum, each term counts the ways to choose a team of $k$ people out of $n$, and then designate a captain and a goalkeeper (who can be the same person). The sum thus counts all the ways to form such teams of sizes ranging from 1 to $n$. Let's now proceed with a different counting method. We distinguish two cases. If the captain and the goalkeeper are the same person, we can start by designating this person ($n$ choices). We then choose the rest of the team, in other words, we choose a subset of the remaining $n-1$ people ($2^{n-1}$ choices). This gives us $n 2^{n-1}$ choices. If the captain and the goalkeeper are distinct, we can start by choosing the captain ($n$ choices), then the goalkeeper ($n-1$ choices), and finally the rest of the team ($2^{n-2}$ choices); this gives us $n(n-1) 2^{n-2}$ choices. Finally, we have

$$n 2^{n-1}+n(n-1) 2^{n-2}=n(n+1) 2^{n-2}$$