You can think of teleportation as the process of sending the state of one qubit from one place to another without having to physically send the qubit itself. Instead, you start with an entangled pair shared between the two locations, and that entanglement is consumed in the process. If you're not already familiar with teleportation, I strongly recommend that you find out about it. It's one of the basic building blocks of many results in quantum information.
Of course, a swap gate is simply sending the state of one qubit from A to B, and another from B to A, which you can therefore do with two teleportations. So, how does this circuit work? Consider two locations, A and B. At each location, there are 3 qubits. So, at A, we have A1, A2 and A3.
Before the start of the main protocol, we create two entangled pairs, one between A2 and B3, and another between B2 and A3. This is the stuff in the red box.
For the main protocol, there is a one-qubit state (unknown) on A1, and another on B1. The aim is to swap them.
The A1 qubit is teleported using the A2-B3 entangled pair, so the state arrives at B3.
The B1 qubit is teleported using the B2-A3 entangled pair, so the state arrives at A3.
A swap is performed from A3 to A1, and another from B3 to B1. Hence, the transmitted states arrive on the specific qubits they were supposed to be on.
You can't quite follow that sequence on the depicted circuit diagram as the final swaps are actually performed a little bit earlier. That just changes which qubits some of the operations are performed on.
Hopefully, you can see that this really is just 2 teleportations run independently to distribute the 2 different quantum states. As such, you can easily generalise it. If you have n parties, and everybody knows in advance where they will be sending their qubit, then, again, each party can manage with just 3 qubits, and they will share two entangled pairs, one with the user they'll be receiving a state from, and one with the user they'll be sending their state to.
Now, if you don't know in advance where each party will want to send their qubit, you could make it so that each location has $n$ qubits: $n-1$ with entangled pairs shared with every other party, and one with the qubit state to be sent. But that is highly inefficient in terms of resources, and it is an interesting question about how little entanglement is actually necessary. I assume this has been answered somewhere, but I don't know where off the top of my head.