It's kind of a phenomenon. Let's get one thing out of the way: it still involves servers, so that word serverless might feel a bit disingenuous. It's actually a new way to pay for and work with servers that, in many cases, is cheaper and easier than buying and managing your own servers.
Here's an empowering example: saving and reading things from a database. That's right! It's possible to have a database and deal with it entirely through front-end code. Technically, the database still exists on a server (someone's else's computer, as the famous sticker says), but you don't need to spin up your own server for it to have a home.
Why bother with doing a database this way? For one thing, it means you can host the rest of your site much more easily. Wanna build a React-powered blog on GitHub pages? You totally can! Wanna use the super fast static file host Netlify to build a Vue powered portfolio site? Do it.
CodePen is a part of this brave new world as well. CodePen can be the home of your front-end code that talks to cloud functions and cloud data. Your serverless host in a serverless world! This very site is built statically with Nunjucks and hosted on CodePen using CodePen Projects.