I know that Run Away allows you to escape no matter what, but Arena Trap prevents escape, so why?
Pokémon with Run Away cannot switch out for another Pokémon, only flee. It does not allow the Pokémon to flee from Trainer battles. Run Away allows the bearer to flee a battle with a wild Pokémon without fail, even if the opposing Pokémon has used a move like Mean Look or Bind, or has an ability such as Arena Trap.
(Sorry, I copied that wrong but you get it)
So Run Away makes Pokemon switch without fail so it'll always switch.
That's just how it works.
To be honest, it's probably to make abilities like Arena Trap and Shadow Tag less overpowered. If you had absolutely no way of bypassing them, Route 13 in Kalos would be more of a hell than it already is.