There's usually more than one server per minigame. You could see it in the url you were redirected to; they had more servers running the more popular minigames. Each minigame has a player limit, so the maximum load on any given minigame server is known (within the bounds of the minecraft sub-superset that makes up that minigame- but usually the minigames are deliberately limited/bounded in how much computation they need, as opposed to vanilla Minecraft). Extra players get sent to the next available server. If there's consistent overflow, at that point you might turn on a whole new server, or change a server's gamemode (I don't know to what degree Hypixel actually did/does this, or how often it's actually necessary).