I am looking for an old scifi story about an athlete that killed an opponent and was sent/escaped to another dimension/planet inhabited by strange human-type creatures with wings. Our hero fights the winged creatures, wins, and saves the girl.
Sounds like Almuric (Wikipedia, ISFDB publication history, ISFDB covers) by Robert E. Howard. Compare Strangeland's description of the novel Almuric (as it was identified in the accepted answer) in this question:
In the early 70s I read what I'm sure was an Ace paper back. Loosely SF in the same way that the Barsoomian tales might be considered SF. A typical "mighty-thewed" hero is magically transported to an alien planet and must save the fair maiden. What I recall most was that unlike John Carter, this hero was not very likable. He lacked the nobility of Carter and was something of a brute. I'm sure it was written during the same golden age of pulps as John Carter. For all I know it might be Edgar Rice Burroughs. I distinctly remember a race of winged humanoids who kidnap the fair heroine and her rescue is what provides the driving force in the novel.
This review tells us that Almuric actually did kill somebody on Earth before fleeing to another world:
Whereas John Carter was the ultimate good guy hero, master sword fighter and clean-cut gentleman, here we find Esau Cairn, the ultimate misanthrope, a man so strong and brutish, he has literally spent his life avoiding sports (or, we may assume, any human contact) for fear he will accidentally kill someone—a situation which isn't helped by his uncontrollable temper. This fellow definitely has issues! It is this combination of temper and brute strength which launches Cairn on his planetary adventure when, having killed a crooked politician, now fleeing one step ahead of the police, he encounters a scientist who—lucky boy—is just looking for someone to test his planetary dohickey out on.
From the same review we learn that Almuric was published after Howard's death, and that some believe it to be a forgery, actually written by Otis Adelbert Kline.