Why can't I see other computers' traffic in the output of tcpdump?

You router and/or switch will only send you packets which are either broadcasted or addressed to your own network card (NIC). That is not the case if you use a hub, which will blindly forward everything it receives to everyone. Having a switch (or a proper router) ensure you can use the whole bandwidth between two NICs, without them beeing spammed by the other NICs' packets.

Imagine computer1 downloads at 100Mb/s from computer2. Computer3 then tries to download at 100Mb/s from computer4. If everyones' packets were to be sent to everyone, computer1/2 would'nt be able to use the whole bandwidth, only a half, and same thing for computer3/4.