Traveler: Thanks, I'll bed down my horse
Innkeeper(?): Well I'll stable him for you
Innkeeper: Have you had your supper yet?
Traveler: No, I could sure use some food
Innkeeper: Well I'll send some up and some fresh water for you to clean up in
Traveler: Thanks, if I can stay awake that long
Innkeeper: Evie (not sure how her name is spelled) you're just in time. This boy is going to stay over with us. Fix him some vittles, will you?
Traveler: I'd sure appreciate it ma'am
Evie: Well I'll send it up as soon as it's ready
Traveler: You said the head of the stairs?
It's hard to know without more context but he's staying the night at this place. Head of the stairs literally means "top of the stairs" like the head is the top of your body. It's hard to know what that last line means. Maybe he's tired and he forgot where his room was. Maybe he's unsure where the room is and it was right beside the top of the stairs.
I haven't double checked accuracy. Maybe someone else can.