This is a very simple one. Think of things that you don't care about and make a list. For me, for example, it would be
*The clothes other people wear by choice.
*The outcome of sporting events or prizes given to movies or poets.
*Other people's taste in movies.
*Almost all of what other people do that doesn't affect me directly.
*Winning arguments with other people.
*How good or bad my handwriting is.
*Almost all comparisons between me and other people.
*The private behavior and public opinions of celebrities.
You could go on and on with this. Now there are things you will care about as well. You can make a list of those things also. Choose only truly significant things, things that you really feel you must care about, like the health and well-being of family members or the state of the republic.
Now the trick is to find a third list of things that you do care about but that you really do not have to care about. Maybe you care about what total strangers think about you, or you care about the quality of their tattoos. You really don't have to. One reason is that caring about these things does nothing good, or prevents no harm, for you or for anyone else. Usually, it will be a thing without any practical consequence, and which you have no direct power over anyway. Your caring about it seems to have some benefit for you, but this is an illusion for the most part. This includes most things that other people think or say about you, and a lot of what you think or say about other people.
You might think not caring is negative, because you should care about yourself and about other people. I agree. You should care. But by moving more things from the care to the don't care column, you will be able to clarify what is really worth caring about. I've gained a lot by stopping a lot of unnecessary preoccupation with things that aren't really that bad, or whose badness is not that significant. For example, I no longer worry that people like the music of Kenny G. I don't need to out-debate other people or convince them they are wrong. While watching my local team play, I want them to win, but the minute the game is over I pretty much forget about it.
If you want to continue to care about something that I wouldn't approve of your caring about, don't worry, because you shouldn't care what I think in the first place. Some of these false cares might seem vital to you, and you will want to cling to them as long as possible. They probably have some function for you. I know I too still care about a lot of things I shouldn't. When letting go of a care you are not becoming indifferent to life, but choosing your battles more wisely by being less ego-driven.
For example, most comparisons to other people have to do with your own ego; so does the drive to convince people your beliefs are correct. Think envy, pride, sloth, wrath: most of that is ego-driven. The other deadly sins are simply wanting too much for one's self: gluttony, greed, and lust. Whatever your framework, I think once you adopt with life-hack you will see that there are at least a few things that your worry does nothing to alleviate.
When should you care about what other people say to you about yourself? I think it would have to be someone who cares about you personally and has your best interest at heart, and is expressing it in a way that conveys that concern, and giving you information on which you can act. So anyone that is insulting you, by definition, does not deserve to be heard.