Your switches run a routing protocol (almost always BPG) and your load balancers all announce the same IP to them. This results in a routing table with many next hops for the LB IP, and traffic is distributed across them using ECMP, typically with some resilient hashing scheme so that a given flow will always go to the same physical LB, even if there are topology changes due to LB or switch failures.

At the edges, your switches or border routers announce the IP to all your peers and transit ISPs.