Everyone should care about politics, at least, for major elections and stuff. Regardless of whether you
care who is elected could and probably will have an effect on you. If say, we're living a country that has one person who makes all the decisions and has no checks and no one to go through (no congress) and for some reason that doesn't actually exist but does for this hypothetical, that person is elected every 5 years. If you don't care about politics and just keep to yourself, never do anything illegal, and suddenly, one of the people who is elected is a crazy bastard who goes around and murders people who eat bread, and you eat bread, that affects you, regardless of whether or not you "cared" about politics. The only way for the people to avoid voting in this really weird guy who murders people who eats bread is to elect someone else who won't murder you for eating bread, and that won't happen if only a select few people vote.
In this situation, let's say 60% of the people love to eat bread, and the other 40% don't like people who eat bread. If the 60% doesn't vote, the 40% will vote in the weird guy who will murder you for eating bread. If the 60% had voted, the outcome would be different and someone else would be in power.
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That only sort of made sense, but I'm tired and rushed, so that's the best argument you're getting from me.