Basically this .tuf file is the Transaction Undo File, which is created when performing log shipping to a server in Standby mode.
So if you ask why standby mode, database recovery is done when the log is restored and this mode also creates a file with the extension .TUF (which is the transaction Undo file on the destination server). In this mode we will be able to access the databases.
Undo file is needed in standby state because while restoring the log backup, uncommited transactions will be recoreded to the undo file and only commited transactions will be written to disk there by making users to read the database. When you restore next tlog backup SQL server will fetch the uncommited transactions from undo file and check with the new tlog backup whether the same is commited or not. If its commited the transactions will be written to disk else it will be stored in undo file until it gets commited or rolledback.
See the .tuf File Location
SELECT backup_destination_directory FROM dbo.log_shipping_secondary