Ok this probably isn't going to make sense, but try and follow me here. The playoffs are like a hot chick. All these other dudes (teams) are battling for her affections all season long. When you or your team loses out, why would you suddenly start rooting for one of the other guys to get in her pants? When your team is out you should call the Playoffs a skank and move on, right?
These people that root for the NL West teams in the Playoffs just because the Padres lost are wrong. You should never root for your rival or another team for that matter. I submit to you that even watching the playoffs when your team is eliminated is a voyeuristic activity. It's like watching a video of your rival foolin' around with the chick that should have been yours.
It's not a case of animosity necessarily. It's like "Ok dude, you're the better man but don't expect me to be celebrating with you later".