The best explanation I can offer is the one by my friend Gojko Adzic, from his old but excellent "Serverless architectures: game-changer or a recycled fad?"
It is serverless the same way WiFi is wireless. At some point, the e-mail I send over WiFi will hit a wire, of course, but I don’t have to pull along a cord with my phone. With serverless technologies, developers no longer have to worry about running a server process that listens on a socket, waits on a queue or dispatches tasks.
There might be even Apache below (it's not for AWS), but it's not your job to configure it, maintain it, and even know about it.
It's not exactly the same as Apache for your web app, as there are multiple pieces of the common serverless app. For example, your business logic and code will be in serverless functions (i.e., AWS Lambda), but the function is fully isolated from the external world, and you need to attach some triggers to it. If you want to make a public (or even private) HTTP API, you'll need some kind of managed API gateway (in AWS, that's Amazon API Gateway). When you receive an HTTP request, you can do different processing inside your function, but you don't have a database (or any permanent storage) inside, so you need another service for that. It might sound complicated, but it's not at all, and it's really powerful, because you can easily give different permissions to different functions in your code. Imagine that two route handlers in your Express app can do only a thing that it's their business logic, i.e. POST /books can just save a book in the database, but it can't read anything, or GET /book/:id can just read a single book, but don't have any additional permissions.
With serverless, you'll get a highly scalable, fully isolated, production-ready app out of the box. And even better, you'll pay per request. For example, my startup infrastructure costs $0.17 for March, and we had 1.5M function executions (and earned much more).
Hope that answers your question. I would recommend checking the free first chapter of our book for more info. In later chapters (chapter 13-15), we show how to use serverless with Express.js, how and why to migrate existing app, and some case studies, exactly because most of the people use Node.js with Express.js at the moment.