Some leecher mods use NickThief, a feature that takes the name of the contacting client, or parts of the name, and sends it back to that client as it's own name in order to exploit it's community boost features, if any are present. eMulePlus is a common example of a community mod. Anti NickThief detects this behavior in other clients.
The purpose of Anti-NickThief is to discover clients using the NickThief feature, and not merely to determine if a client has the same name. Having two clients with the same name is not harmful, though it can be annoying. But a client using NickThief will in most cases employ other bad features against other clients, such as enforcing a very low sessionupload, fake ranks or even no ratio or 0 upload. That's why one probably wants to detect and ban them.
Detection of NickThief is simple. AntiNickThief sends a random string to each user, such as "username [abcde]". If the contacted client will adapt that string to its name and send it back, it can safely presume that the client is employing NickThief.
Note that there is also an Anti-AntiNickThief bad feature which attempts to detect clients using AntiNickThief (There is no honor amongst thieves!). Because of this, newer versions of the Anti-NickThief send 2 tags and/or tags of random length to make it as hard as possible to escape the bans it imposes.