I guess that expression is used when someone is talking about something that he/she doesn't want to care about, I mean, 'Why do I have to care about? I don't want.' Is my guess correct?
What do I care?
In spoken discourse, this is almost certainly going to be spoken as "What do I care?", where the speaker emphasizes I to make a distinction between the speaker and the other party. Using this phrase shows your indifference to what the other party is interested in. I believe it is meant to be ironic, but using this phrase will often come across as mean-spirited.
-- There is a discount on sliced bread at the supermarket!
-- What do I care? (= I understand you have an interest in that, but to me it is of little or no consequence.)
A similar phrase is "What does he know?", used to shows a contrast in knowledge in an ironic way.
-- I think we should go that way.
-- What do you know? We normally end up lost when we follow your advice.
-- This looks like genuine gold. But what do I know? I'm no expert.