As someone who struggles intensely with anxiety in general, especially that of the social type, I find it easy to relate to others in a similar boat. There's nothing fun about it, and those who don't suffer from it don't always understand what the big deal is.
Of course, this is an issue that needs to be pushed more into the forefront of people's minds as something to consider when talking to somebody new.
You never know who is dealing with anxiety.
Some of us have grown to be good at hiding it from those we don't want asking about it, and others are pretty transparent about it. In many cases, you can almost tell just by looking at someone if they are dealing with this problem, but you would be surprised sometimes at who around you is actually finding themselves constrained by the swarm of people going about life all around them.
Human interaction is honestly quite an amazing thing to experience when all parties are on an even playing field, and despite being a major introvert with this anxiety, it's one of my favorite things. This all comes down to the people you choose to spend your time with, and the amount of understanding they have of what you're going through.
The more effort you make to understand others, the easier it will be for them to enjoy your company.
Getting comfortable with others can be a major hassle when you find general communication with others stressful. It's far too easy to freak ourselves out overthinking every interaction with others to the point we actively avoid them.
When others are able to understand your mindset of always worrying what others think of you, whether they also have anxiety or not, it makes it a lot easier to open up to them. This is the quickest way to actually befriend someone who is fearful of interaction, despite craving a connection.
The smallest things to you can be much bigger to anxious people.
Overthinking is the name of the game, and having anxiety makes us the unenthusiastic players.
There are many things a lot of people won't think twice about: tone of voice, leaving text conversations unfinished/leaving on read, going multiple days without communication, or even simple facial expressions you make when talking. This is only scratching the surface, but it's impossible for us to not notice every detail before sitting alone later thinking it all over repeatedly until we find ourselves in an unhealthy cycle.
The chain can be broken, with immense willpower or if the situation is nullified by whoever unintentionally caused it (if they clarify what they meant, return to continue your text conversation, etc.). You can't really avoid these completely but we still ask people to make the effort to understand why we overthink about these likely small things.
Don't be scared to get close to someone with anxiety!
More often than not, we want a deep connection just as much as anybody else. It's harder to express these feelings when we're battling our own intrusive thoughts, but we will put in the effort for those willing to do the same back. This doesn't mean it's going to be a smooth ride all along the way, but it will certainly get better the more you care about them.
Simply allow yourself to ease into a conversation. Don't rush anything, and definitely don't pressure them. Let them know you're there and willing to listen, and with time, they will become comfortable enough to open up. It's all a matter of understanding and patience.