With BGP you "own" public IPs and they are in an ASN number.
Routers register their ASN numbers, and then advertise their public IPs as prefixes to other ASNs it has configured as "neighbours", think of it like telling other routers what it's own IPs are and that they should put it in an ACL and then keep advertising that onto other routers, so that things can make their way across geography to the destination. Routers that know the neighbors of other routers, etc.. can figure out the best path to the destination.
If you advertise your prefix wrong, it can cause major routing issues, either things go down and there's no route to it, or different routers know an IP as a different ASN due to a wrongly configured router advertising the same IP as another, it can cause loops and things like that which aren't quickly sorted.
This is more small scale though, in the larger picture for an ISP they are bringing datacenters offline and things like that, then they lose access to internal tools and it becomes very complicated to get on site and start connecting locally to do updates.
That's my take of it from someone who did some work for a startup that had a /23 block of public IPs for eBGP. The benefit in this case is that workstations that were critical to operation connected to things on their public eBGP IP. If there was a network blip or outage on ISP1, BGP would failover to ISP2, but the workstation still had the same public IP and things like TCP negotiations could continue.