When you care to do something, you want to do something; be willing to do something.
To be interested in something; inclined to do something.
(Opposite) To NOT care to do something means that you do not want to do something; have no need of something; have no wish to do something.
1. Mr. President, would you care to comment on the recent issue on racism?
2. I don’t really care to see James right now.
3. Would you care to join me for a drink tomorrow night?
4. I don’t care for compliments at a time like this!
don’t care, don’t give a damn, could care less, emotionless, flat, have no need of something, have no wish to do something, have no use for, indifferent, unaffiliated, unattached, uncommitted, unconcerned, uninvolved