Or have someone else deploy it for you: https://www.vultr.com/docs/configuring-bgp-on-vultr/ | https://netactuate.com/anycast/
All big cloud providers have anycast load balancer offerings. In my experience, Fly's load balancer is wayy more simpler than those to use (because it is the only way to load balance, like with Cloudflare).
Note that in the case of fly you don't directly get traffic to an anycast IP but through a reverse proxy.
I could imagine that they replicate connection states to all pops so that traffic that arrives in the wrong pop can be rerouted.
In the past people were concerned more about route flapping disrupting TCP connections. It seems that people aren't worried about that anymore. Google does it.
Aha, that brings up technical articles, thanks. TFA does mention it multiple times so mea culpa.