Imagine that your Internet service is just an Ethernet cable that goes to a router in a datacenter. This is Apple offering to plug their servers into that router. Now you can get to their servers without going out to "the Internet" via a Tier 2 or Tier 1 ISP. That is where the word "internet" comes from -- interconnected networks. More connections is more internetting.
This is all super common. Many big companies are happy to peer with small ISPs if they're already in the same building.
Edit to add: The edge cache thing that this article is about is similar, but not quite the same as what I'm describing. Instead of connecting you to their network, they just put some of their servers in the same datacenter as your network. Even less latency!