With ordinary unicast, an IP address belongs to one machine in one place. With anycast, servers in many data centres announce the same prefix over BGP, and the routing system delivers each packet to whichever location is closest in routing terms. The user sees a single address; which site answers depends on where they are.
DNS is the classic use: the root servers, TLD nameservers, large public resolvers and most managed DNS providers are anycast, which gives low latency everywhere and absorbs attacks by spreading them across sites. CDNs use it to bring content close to users. If one location fails, it withdraws its announcement and traffic flows to the next nearest.
For lookups this has a practical consequence: an anycast address has no single physical location. A geolocation database shows one place, usually where the block is registered, while users on different continents are in fact served from different cities.