When you send traffic to an IP address, your router must know where to send it to. In consumer routers, you usually just have a default route for the whole internet towards your ISP's gateway, but the ISP must also maintain a routing table to know where the packet must go next, and BGP is the protocol most commonly used to advertise where each network resides.
Every ISP could in theory maintain the internet routing tables by just having a set of static routes, but considering that the IPv4 internet routing table consists of over half a million entries at the moment (and it's growing), this is practically impossible. Instead, each ISP (or "Autonomous System") peers with others to advertise their routes to the network blocks assigned to them, and often the routes they receive from other peers.
When you have multiple peers like this, the advertised routes contain as metadata the IANA assigned global AS numbers of each participant in the network up to the endpoint (the "AS path"), thus allowing routers to calculate the shortest route in case two peers advertise routes to the same network block. This also gives the internet its self-healing property, since if one router goes down, you usually have an alternate path available.
The BGP "communities" mentioned in the article are a way for peers to signal additional information about their routes. For example, an ISP can publicly claim that they will not forward traffic to routes advertised with a certain community, allowing a network operator to tell their transit ISP to drop DoS traffic to a subset of their addresses before it reaches their network, helping protect customers not under attack.
The Internet is a set of interconnected networks. These are "internetworked" together.
BGP is the protocol, border gateway protocol, which lets those individual networks know how to find one another.
The individual networks are known as "autonomous systems". This is a single uniform zone of autonomous control, under a single authority (more-or-less -- there are exceptions to everything). An Autonomous System (AS) is identified by an Autonomous System Number (ASN). It's comprised of a number of contiguous network blocks (go look up The CIDR Report if you want to see these). And it finds its neighbours and peers by way of BGP, as previously noted.
I'll second the recommendation for Halabi's book, which remains the definitive text.
Good question, and though this isn't something that turns up every day, it's what's underlying your use of the Net.
Imagine you had to deliver a package to a university but only through your direct friends. You'd have to hand it off to somebody who goes to that university or somebody who knows somebody(or knows somebody, etc) that goes to that university.
BGP helps routers maintain the list of their friends(other directly connected routers) that can get packages(packets) to remote locations(networks).
It's probably not the most up-to-date treatment of BGP and Autonomous Systems, but it's pretty good for learning the basics.
I'm assuming "many more" networked devices being added to the global network will eventually result in more traffic and more ASs being connected.
Long answer is, actually these changes already happened quite a while ago.
Most of the time, we tend to forget IPv6 is in fact more or less 20 years old now.
Support for IPv6 in OSPF (OSPFv3) was introduced in 2008 and support for IPv6 in BGP was added in 2007 (MP-BGP).
AS numbers encoding was also changed in 2007 (32bits ASN) to support all those new connected system.