In many sports players "line up" in a certain position, often with a small space betwee facing players. One is "offside" if they are over the line into the other teams positions.
X | Y
X | Y
...| XY (the X player is "offside")
If two people are friends (chummy) then in any dispute with some other person, they would be 'on your side' out of friendship.
If the person is 'offside' with you, it means she is not feeling very friendly towards you, the implication being there has been some sort of falling out but it is expected to be temporary - "She's a bit offside with me at the moment so I'll wait a day or two before I ask her about it (that is, after she's got over being angry/offended/put out(=upset and annoyed) by something you have done.)