Is this the first time you've heard the term serverless? It means your code is plug & play, you don't have to manage the infra (I.e. a server) yourself.
I understand the concept of serverless as in AWS Lambda, but my first impression reading the title is serverless as in no backend. That would be a lot more amazing.
The "API server" only runs when requests come in. Otherwise Modal scales everything down to 0, and you don't pay for anything. It's similar to putting Lambda behind an API gateway.
"serverless" is a throwback to 2013 a decade ago when it was a novelty to not know where your servers were or what they were made of, some people never move on. but in this case, this is a company advertising their serverless architecture for that specific crowd.
this is distinct from "client side" where no servers are involved, aside from maybe something that serves a static web page.