How to make them understand?

Obviously I understand that your mother doesn't understand, and that her point of view may be biased because she hasn't dealt with what you're feeling. Try maybe bringing up some information on social anxiety and discreetly brining it up around the house. hint it. Also you may want to get professional help if you want to work around your social anxiety and be the best you can. Also, why is she so against you being homeschooled? Try to make her see that some people are just different and do-not mesh well with others I'm always free to talk :)

Unfortunately, being around people is a fact of life. In order to function on any level you have speak to people on a day to day basis. Dropping out of school would be a really bad idea. It might not feel like it now, but you will really miss out.
I suffered from social anxiety and shied away from people. I ended up with pretty severe agoraphobia (basically, not being able to go outside) which took two years to battle. It can start by avoiding certain situations (such as school) but can soon affect your entire life- I couldn't walk 5 minutes to my local store without panicking.
Your mother is right, you do need to be around people. But you obviously need some support. Your mom might think that you're just being awkward, why don't you sit her down and tell her how you really feel? Instead of asking her to drop out of school, why don't you ask if you can go and speak to a doctor or therapist? Try speaking to a favorite teacher or your school counselor, they genuinely care about your wellbeing (most of them, anyway).
I hope I've been somewhat helpful, and I hope you find an answer.

Endure. Have you ever thought that your Mom probably believes that you can overcome your anxiety? There's nothing wrong with trying to be brave and fight. Of course it's much better if you can seek professional help like they say. This is just my opinion but I think home schooling is just running away from your problem. Imagine if it get's worse, you may not want to go out of the house. Next, you wouldn't want to go out of your room. You are limiting yourself.

I know what you mean. I had some issues but my mum kept telling me that I was imagining it and that I should fight. After a while, I gave up and just started enduring. Turns out, what she said was true. I overcame my issues and at the same time, tried my best to seek advice from her. In your case, like alot of the others have said, interacting with people is NOT something you can avoid in life. Remember, no one is trying to look for moments when you mess up. Try to be confident as its key to succeeding in such situations. You need to try your best to fight and make sure you believe in yourself. you can surely overcome this :) However, if you feel like its too much, try to talk to your mom seriously about this. Dont push towards homeschooling but try to suggest getting therapy or any minor help possible for this. Hope you work this out! Goodluck :)