Basically, introverts recharge by being alone while extroverts recharge by hanging out with people. Whatever you extroverts feel when you have been away from people is the same feeling we introverts feel when we are always surrounded by people. It's very draining, causing us not to function well. Additionally, the world we live in seems to be made for extroverts (like school, work environment, etc.). Introverts need to force themselves to get out of their caves so that they can be "successful" in this world. That itself is already very tiring. We value solitude very much because that's when we are at peace, that's the best time to be in our flow state. Yes, we do need to hangout sometimes, but we do it only with people we truly enjoy being around with. Personally, I enjoy being around people who understands that my silence doesn't mean I am mad at you or bored being with you. I just prefer to listen and observe rather than talking. Majority of my closest friends are introverts as well. When we're hanging out together, we can be as loud as you extroverts because we're comfortable with each other. We don't like being forced to hangout with others. That'll just cause stress and anxiety.

If you really want your introverted brother to hangout outside, please don't force him. Let him do what he wants and find his own tribe. If you truly care about him and want him to open up to you, you need to be extra patient with him. Spend time with him doing what he loves. If he loves video games, then play with him. Bring him somewhere he's interested in. For example, if he likes anime, then bring him to an anime convention. He'll meet his fellow geeks their and enjoy his time. After that, let him have his alone time because his energy would be drained after being in a crowd, socializing.