Why as a player should I care about resonance realms? I understand that the actual character would be totally into hanging out in them, but from a gameplay point of view, why should I care?
Unfortunately this is mostly in relation to how your GM runs the game. The resonance areas are like the astral realm: If you GM doesn't really support it, they mostly become flavor for the setting. However, when used you have alternate routes to find info and wreak havoc... And so does the environment. The primary downside is spotlight time. When you go somewhere the other players can't (unless your GM lets them control your pets). That means the game is focused solely on you. Especially considering that you usually get extra passes per round even when the GM straddles the two (or three) worlds.
If this question is geared towards how you would make your character, you should talk to the GM about the setting and how they will use your technomancy.
EDIT1: You may want to make a character less inclined to use the realm as a backup and explain to your GM that this is merely an alternate build in case they don't support the way your character is (expensively) built.
As a gameplay benefit, I read somewhere in the Unwired book that "Resonance never forgets", and a Technomancer accessing its Realms could retrieve any deleted file (assuming they were in a Matrix-connected hard drive, I think). Your character could also meet there precious allies (and deadly foes).