Explanation
One difference of serverless hosting is that a third party manages the hosting for you. However, you can also outsource traditional hosting, so what makes serverless hosting truly unique is that your files are stored and ran in the cloud. Of course, the cloud still needs servers to run, but they technically have nothing to do with your website. Something different about how your code works is that it's not running all the time. The service that makes this happen is called functions-as-a-service, and the process of the functions running your code is called event-driven execution.
Why Use
First of all, you don't need to manage anything related to servers. Someone who knows what they're doing still needs to manage it, though, which is why serverless hosting should be done through a third party. Also, since the cloud is more dynamic, it can handle any amount of traffic at any time. Besides meaning that your site would never really go down, dynamic traffic handling allows you to only pay for what you use. Like I said before, the code doesn't run all the time, so you wouldn't even be paying for those periods.
Amazon Web Services
Google Cloud
Summary
For the most part, serverless sounds great, but I do see one immediate problem versus in-house hosting. When working on the one app at my company made for the outside world, I have had to change some things on the server. While you'd still be able to communicate with your host and have them change things, I'd prefer to avoid that hassle and fix it myself even if it takes longer to figure out.