Basically social anxiety makes chatting and talking very difficult. The anxiety part makes you see potential dangers where there aren't any, such as "what if the thing I say comes across as rude, or over-friendly, or stupid, or offensive" and in addition, you tend to be more sensitive to emotions so rejection or ridicule would hurt a lot more. It doesn't mean talking is impossible or that socially anxious people dislike talk or company in itself, it's just that it's difficult. This is a disorder, it's your brain being wired differently so while you can work on it, you can't just snap out of it or grow a thicker skin.

Not belonging isn't necessarily true, everyone belongs somewhere and can make friends, but it's easy to perceive it that way if you fall into the trap of thinking social interaction is an impossibility for you. Having that much struggle to simply chat to someone can be discouraging. Honestly it can be a bad habit sometimes, it's unhealthy to get too deep into an "everyone hates me" pool of self pity, you tend to push away those who care. It's not always the worst thing to point this out (just in a nice way, not an angry one). Even if they don't seem to listen, they can understand it.

The best way to handle the situation for talking can be to accept they won't often initiate conversation. Nothing wrong with poking to chat as long as it's not constant or demanding, it can often be nice. Asking questions or bringing up a subject of mutual interest can help since that's a lot easier than being expected to contribute your own topic or flow of conversation. For example I'm terrible with chatting about day to day things, but I could ramble about something ingame all day. Quite a few with autism have that trait of very specific focused areas of interest. If they're quiet or drift away it doesn't mean they're being rude or dislike talking to you.