As someone who still checks in to places using Swarm (FKA Foursquare), the novelty of paperplanes.world speaks to me. The point? To "throw and catch paper planes with people around the world". I suppose it serves a similar purpose as a message in a bottle or putting notes in balloons and waiting to see if someone responds. Communicating with strangers is still a thing.
The desktop site shows a "real-time" visualization of where planes are being thrown. You can't actually interact with the desktop version of the site, though. That is done through your smartphone. From there you can create your own planes and track them to see where they've been. You can also catch other planes, add your stamp and send them back out into the world. Stamps only show dates and locations, so the whole experience is relatively anonymous.
There's also the added benefit of knowing your plane will soar every time. That does't always happen in real life.