Hello, I see you posted this a month ago, hopefully you return. When you ask what someone's connection to the anime or Lain herself, I presume you're asking what draws a person to the show.
For me, it clicked almost immediately while watching. I saw a couple memes with Lain in them and I thought she looked adorable, so I decided to check it out. I remember finding it so strange, but intriguing at the same time. It was nothing like anything I'd seen before and I'm not just talking about anime. It was such a shock when I found out it came out in the 90s and wasn't as popular as other forms of media like EVA and Akira. I remember trying to individually get close friends and family members to watch the show and realize it's elegance like I did, but it clicked with none of them and even die hard anime watchers I know in real life had never heard of it. Even watching it with other people made the show feel bad, however, this only solidified how special it was to me; how special it made me feel. I realized that I, while not being special or anything, saw things like this differently than most people. It felt like I was understood by a form of media, but more importantly, it felt like I was understood by Lain herself. I believe anyone can see Lain in themselves in some form or fashion, but it takes a certain kind of person to love Lain. Over the years after watching and rewatching a few times, I happened upon the other forms of Lain like the short manga and PS1 game. Shortly thereafter I found little pockets of the internet like mebious, arisuchan and now this place as well! Of course this would be the fanbase, it wouldn't be right if it was anything else. The amount of love and admiration for Lain I've seen is incredible. It brings out the most creative people I've seen for a long time. The people that got it, like I did, do really exist in the world. I'm not alone! Let's all love Lain!
My connection is being a huge fan of The Big O and Digimon Tamers back when they were first airing in the West, finding out later that they were both written by the same guy, and eventually checking out the other shows he wrote for. I guess I'm more of a Konaka fan than strictly a Lain fan.
What blows my mind about a lot of Konaka's stuff is how he kind of predicted the state of the modern Internet well in advance. Like, it's hard to look at Eiri and his worshippers and not ponder the unhealthy parasocial relationships that so many lonely people have with e-celebs and Twitch streamers these days. It's hard to watch the D-Reaper feed off Juri's misery and not draw comparisons to the social media monopolies who have built so much of their business model on doing just that. I haven't found any other writer who's tackled these sorts of big issues so elegantly without being overly corny/preachy about it.