it just means they deal with some obvious and non-obvious problems having to do with being A Big Deal. (there's only a little bragging involved.) the most usual example is scaling up. say you have a website which you run on a server, and it's humming along serving users. everything's great. but then you get popular, and your one lil server can no longer keep up. So you swap that in for a bigger server and everything's great again. but then you get even more popular. and suddenly there don't exist bigger computers. so then you split up the programs on that one computer so that you can use two big ones and serve more people than you could before with one. and then three servers. and so on.

Someone who's dealt with websites at scale before is going to be able to skip the first 20 of that process, so if you're a startup dreaming to get big, you find someone that's worked "at scale" eg Amazon or Google to design and build your systems and avoid some pain points like the website going down because the system can't keep up.