While "Just put yourself out there!" is technically valid advice and ultimately the crux of the solution, it's obviously a little more complicated on an internal level.
First make sure to clearly define your goal: many people view social confidence as being aggressively outgoing in terms of communication with others, which is not the case for a huge amount of people. Realize that social confidence is feeling like your contributions are valuable, not necessarily that you're constantly contributing. Feeling like you always have to be "on" is really stressful and exacerbates the anxiety you want to overcome. It's not necessary.
Further, what I found really helped me is that whenever you're having anxiety about someone judging you, try to imagine in your mind how you would react in the given person's shoes. Would you be mad if someone asked you a favor? Do you go out of your way to judge the new people you're meeting? Probably not, and such is the case with 99% of people. People want there to be a cohesive social environment, they're not out to get you.
Try to train your brain to fully appreciate that what confident people talk about isn't generally objectively more interesting than what you think about. It's all about the presentation. Your ideas are just as good, and if you present them with the same level of confidence and enthusiasm as confident people do, you will be received in a similar way. You'll start to notice very quickly that if you talk like you know what you're talking about and you sound passionate about it, even if what you're saying isn't super awesome, people will be drawn to you.
If you feel like you don't have anything to say, ask them a question about themselves. People love being asked about themselves, and if you give genuinely interested feedback or compliments, they will love you.
Try to practice in an alcohol-lubricated environment. Not trashed, just something to take the nerve away. You'll be less anxious, they'll be more receptive, and you'll notice just how easy it is. The only hard part about it is in your mind, the actual steps of emulating confidence are ultimately very simple very quickly, but you have to get the ball rolling. Eventually as you realize that your faking it is working, it'll become more natural.
The first hurdle is the hardest.