World of Warcraft Classic exploded in popularity as soon as it dropped which caused massive server issues for Blizzard. The company is looking to both expand the capacity and solve those issues by offering free server transfers to players.
World of Warcraft Classic's servers quickly filled up but some of them were not as populated initially. To bring the population up in those servers and lower the numbers in the overloaded ones, Blizzard will be offering players an option to transfer for free.
Mind you, not all servers will get this option as that would defeat the purpose. The official post on WoW forums only stated the transfers will be available for "select Classic WoW realms in the coming days" but it's pretty obvious that it's the full ones.
If you happen to be on one of those realms, an option on the character selection screen will pop up to allow you to go somewhere less crowded. Better yet, Blizzard will add the same option for the queue screen so you will not need to wait half a decade to get in, just to transfer somewhere else.
While anyone can transfer, there are some in-game that can cause players to be unable to do so, at least temporarily. In case a character is a guild master, has mail or active auction listings, they will not be able to go a different realm.
These are mostly easy to solve as active auction listings can just be taken down and mail read or deleted. The guild master characters may have a harder time though, especially if they spent a lot of it rounding up a community they are now the head of.
Blizzard commenced the first round of realm transfers at the time of writing and they are quite specific about from and to which realms players can transfer. You can check the full list out on the by Kaivax, the community manager.