What does this do?
TransferAPI is used for moving players to another server on a BungeeCord network. There is no advantage to non-developers currently.
I'm a server owner, my developer told me to download this?
Awesome, all your developer needs this for is to connect players to different servers across your network.
I'm a developer, how do I use this?
Developer Documentation: http://dev.bukkit.org/bukkit-plugins/transferapi/pages/developer-documentation/