Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected users of the system (n^2).
Robert M. Metcalfe, the inventor of Ethernet, originally meant it to apply to devices on a network that could communicate with each other. It isn't sufficient to be on the same network if they speak incompatible protocols. It isn't sufficient to speak compatible protocols if they aren't connected.
A more plainspoken way to state Metcalfe's law is that every one new user added to a network makes the network more than one unit more useful.
A more simple way to understand this law is: "The first person to buy a fax machine was a fool." Imagine how useless it would be to be the only person in the world with a fax machine. You can't send anyone a fax because nobody else owns a fax machine. When two people owned fax machines the utility or usefulness became a lot more, assuming those two people needed to communicate. When 100 people owned fax machines it was more than 50 times more useful than when two people owned them because now 99 people all had 99 other people that could communicate: 9801 pairs. Maybe not all 9801 pairs would be used but the network had a lot more potential than when only a few people had fax machines.
When Metcalfe invented Ethernet very few computers were connected to each other. Communicating between computers usually meant writing data to media such as tape or a "disk pack" and physically moving the media to the other computer. This was often done even if the computers were right next to each other. Think about all the old movies you've seen where computers have tape drives spinning big loops of magnetic tapes. That's how data got between computers.
Whether Metcalfe's law is exaggerated or misapplied to other things, the general point is correct: Linear growth in the number of users creates superlinear growth in the network's usefulness.
I think every sysadmin should understand this law. I think we "get it" as far as the literal sense: We get that more connected computers is more useful. We gain huge satisfaction when we add a device to our network; especially if it is one that previously couldn't connect such as when WiFi is added to a home thermostat, television, or phone. We get that more compatibility within our network is more useful. We are frustrated when two software systems can not talk with each other; we get huge satisfaction when vendors provide standard interfaces, APIs, and file formats so that more things are compatible.
On the social level Metcalfe's law applies as well. If you belong to a local or national user group gaining more members isn't just a matter of pride. Every additional member adds to the potential utility of the group. Every active member adds utility superlinearly. If you are a member of such a group, getting your friends and co-workers to join (or getting current members to be active participants) benefits you and all other members more than you'd think.