Hoarse is an adjective which means "sounding harsh, especially because of a sore throat".
The pony is explaining to the bartender that perhaps it was difficult to hear him because he was hoarse, i.e. his voice sounded harsh.
"I'm a little hoarse" sounds like "I'm a little horse". So it's a play on the words hoarse/horse - because he is a pony, so he is a "little horse".