A person’s “side” in this sense, is their opinion, position, or the way they are viewing something. When 2 people are having a debate or an argument, they have different viewpoints, or different “sides”. People who are listening to the conversation but not directly involved probably agree with one of the people more than the other. If you agree and think the same way as some person regarding some issue, you are on their “side”.
It’s kind of like being on their team.
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