People are rude because they are incapable of articulating their differing opinions in a well thought out manner or they are insecure in their opinions. They are rude online because they are so used to everyone else being rude and they want to build a following of support to help reduce their insecurities and doubts.
My grandfather was a people person, but he was also a debater. He would debate with just about anyone on any topic and he would go on and on about it. My father was even more opinionated than my grandfather, but I noticed my father rarely got into debates with anyone including family. Whenever he disagreed with someone, he usually stated his point, responded to the first response he received, then he let it go.
When I was young, I was more like my grandfather in that I would debate endlessly. I asked my father why he tended to let everything go or even worse, let other people have the last word in debates. Beyond his normal "You can't fix stupid" saying, he said that debating to win is a lost cause. You should debate to inform, not to win.
I firmly believe in that philosophy now which is why when I get into a debate with someone online, I typically do not respond more than 2 times to the same person. There are some exceptions, but they are rare.
I do the same in the offline world as well. I don't have the "I must prove I'm right" mindset. If I believe I am right, that is all that I need. I do not need confirmation of it from anyone else nor do I have even a remote desire to convince others they are wrong. I believe you believe what you believe and the only person who can convince you to change what you believe is yourself.