Personally I consider this topic one of the most difficult as well as one of the most clarifying insights I ever came across.
Difficult, because it’s ‘obviously’ much easier to put blame on someone else. It’s much easier to point a disapproving finger at someone else, then to look critically to your own functioning and character.
It’s not easy, but the reward is incredible. By taken responsibility for your own emotions, you no longer give the power over those emotions to someone else.
In principle it means everybody in your life is a mirror to you. Even people you don’t know personally. If someone irritates you, (s)he symbolizes that part in yourself you don’t want to look at.
So it all starts with the question: what is it (exactly!) that I find so irritating.

Do you consider someone to be sanctimonious? Saying one thing, but doing something else? Watch you life honestly and scrupulously. Changes are you’ll find something sanctimonious going on there, too. Maybe you’re telling someone one thing, while meaning something completely different?

Are you annoyed when someone imposes his/her own views, without any respect for dissidents? Watch you life honestly and scrupulously. Is there a part of you that wants to be right, or wants to get its way, or win, no matter who’s gonna hurt?

Are you angry when you read about someone who killed a human being in a blaze of anger? Watch you life honestly and scrupulously. Are your reactions dictated by anger, sometimes? How short is your own fuse?

By using your resentment as a mirror to become aware who you are, you will remove that reflection away from the other person. That person will have no more influence, no more power, over your emotions.
If you start working with the true source of your annoyance, by recognizing and acknowledging that part in yourself, no one will be able to rattle your cage anymore.

Do you feel powerless? Do you feel you have no influence in this current society? Let this insight and its consequences sink in and see your own strength, your own power and your own responsibility in this role play.
It’s not easy. But it is enlightening!

Building on,