"Running a datacenter" isn't something that inherently needs its own IP range. You could run a datacenter only in private address space and NAT to the internet (though this wouldn't really be very practical) or even run in private space. My point is that having a lot of computers doesn't _necessarily_ require getting public IPs or ASNs.
You need public addresses if you want to be publicly reachable. If you're small you might find an ISP — let's say ISP Inc — that will give you an internet connection and allocate some part of the address space it in turn has been allocated. There's no need for BGP or dedicated IPs here — the rest of the internet already knows how to reach the range(s) of addresses assigned to ISP Inc, and from there ISP Inc's own network takes care of sending the traffic down the pipe to you.
If you need a lot more addresses, or you want to use more than one ISP (either of which could apply to WhatsApp's case), you probably need to get a dedicated assignment and an ASN. The ASN means you're an "autonomous system" — not a carved out bit of someone else's network. You go to your local registrar and convince them you need some addresses, and then you go to your ISP(s) and ask them for "transit", which means they'll connect you to their network and route to and from the rest of the internet for you. But since you're now using your own addresses it's not as simple as above, where your IPs were part of your ISP's range and everyone else just sent their traffic to your ISP for them to route to you. Your IPs are yours now, and until you tell the rest of the internet how to find you nothing will work. To make this work you have to get your ISP(s) to tell the internet "hey, y'all want to send traffic to Mr Weasel LLC? I can handle it for you". This is called "announcing" a prefix (IP range) and BGP is the mechanism through which all the routers on the internet propagate announcements detailing who is providing connectivity to who.
Conceptually this stuff is quite straightforward but in practice it can be mindbending. As well as transit there are peering arrangements (where Mr Weasel can arrange with Netflix to swap traffic over a dedicated physical connection, rather than their respective ISPs) and hella complicated traffic engineering schemes linking the big content providers to the internet. For instance, Facebook has thousands of transit and peering connections with other ISPs and providers and deciding which path is optimal for any given situation requires very complicated policies and management.