Loki actually has quite a bit of experience patching up idiots. You gain it quickly by growing up with someone like Thor. "This will hurt. Immensely," he warns. His attention first is on the arrow head. Luckily it can just be screwed off and discarded. He grabs Clint's hand on the uninjured side, puts a wad of cloth in it then guides it to his arm. Next he holds out a piece of wood Clint can bite down on. He waits to see if he'll accept it before doing more.