Online interactions are different to real life. In a tech forum many answers to questions are binary, right or wrong, 'black or white'. And that can be quite comforting to some people. But that isn't always the case in regular human interactions, and even when you are discussing a situation that IS, not everybody has the same thought process. In your question you talk about trying to "make a person understand", and it sounds like you are unsuccessfully trying to port your experience answering questions online to real-life.
The very fact you are here, trying to figure out why you react this way, shows that you do care what other people think of you and even perhaps how they feel. So I don't think the fact you want people to just accept what you say first time, every time, means you are arrogant.
You need first to analyse your feelings. Perhaps you feel that if you have gone to great lengths to learn something, someone not accepting what you tell them first time is challenging your own thought process? Or perhaps you see it as an insult to your intelligence? Or maybe the challenge is taking away the comfort you find in being right about something? Whatever it is, it isn't worth getting angry about.
Remember also that everybody has a different thought process, and while you can sometimes tell somebody a straight answer to a question (eg 'it's 42') other times you have to lead them down their own thought process, so they can understand how and why that is the answer, so they can have the same confidence in it that you do, and perhaps even gain some of that comfort that you may get from knowing that you're right.
Try to see imparting information to somebody as a worthwhile challenge rather than a chore. Rather than give a quick answer, try replying to complex questions politely with questions of your own that start them on the thought path. If they really want/need to know, they will get that you are teaching/training them and respond favourably. If they don't like your new Yoda-like way of answering questions, they'll give up and start asking someone else, so it is win-win.
When you DO have to give a direct answer to something, you can always say something like "in my experience..." which gives them the option of listening to you, or not. And if the situation does still arise where someone is just not accepting what you say - remember that there is only one person you can successfully control, and that is yourself. You are not responsible if somebody gets something wrong because they didn't listen to you!