battered = damaged, beaten felt = a type of material, usually made of wool that's pressed together instead of woven draggled (more often "bedraggled") = dirty and wet, usually from having been dragged on the ground, or through mud

"Battered" means beat-up, ragged, or worn down. "Felt" is a kind of material, a soft kind of cloth. The hat is made of felt. So another way of saying the first sentence is, "he had kept on his beat-up old cloth hat". "Draggled" is not a common English word. Usually in English we would say "bedraggled". "Bedraggled" means his mustache is stringy, worn, maybe dirty. He does not take good care of his mustache.

hi guy why did you say it was a hard and confuse sentence for you/ ,