* What fans influence: subconscious referee biases (not wanting to be booed for a close call), heckling that slightly impacts players’ focus, objects being waved in their line of sight, noise levels rising at inopportune times, etc.
* Travel/comfort: the home team players sleep in their own custom-tailored sleeping environments (these can include stuff like oxygen tents, not just nice mattresses), don’t spend time encased in a plane / train / bus environment not designed for optimizing future athletic performance, and don’t have travel stresses and logistical headaches. Also for those inclined, the local club / strip club / groupie scene doesn’t hold the same mysterious allure on a random Tuesday night as it might for the lad from out of town.
The second set of factors remains even when fans aren’t allowed inside the arena.