With PAAS services, you build you ruby on rails/django/spring/express app, and PAAS provider takes care of provisioning this app on its infrastructure. You still have to make your application scalable, and specify to PAAS provider how to scale your app (like you need to provide auto-scaling of database, app servers, etc.
In serverless programming. you split your application into multiple functions (some functionality that needs a few minutes to complete, but not long), each of which will be triggered by some event. Events could be an http request, or message queue event. So, you need to redesign your app to be reactive (event-driven) - and a collections of functions. For example, an http request will trigger a function A, which may emit a message-queue event, which will trigger another function B.
Then you just specify this triggering event to function mapping to your provider, which takes care of scaling each function dynamically, based on the load.