When you uploaded your PHP script, you uploaded it to a single server, and it ran on a single server. Unless your provider were doing some kind of load balancing (doubt it).
With serverless you upload your 'script' to the 'cloud' and then it will run on any number of servers, usually assigned to one when it dry-runs. Then it'll unload from the server afterwards. And therefore unless you have a bottleneck (like a database server) then you can essentially scale as far as the cloud will let you.
At least with Mediatemple’s shared hosting, they had a “grid” service that would host your site on a cluster of servers and claimed to be able to handle large bursts of traffic.
It worked because each page request ran a complete cgi php script to generate the page view and then terminated. There was no long running server.
