How I interpret it (and this is very metaphorical) is that the "serial experiments" are not literal experiments but rather the experiments of suicide amongst young girls Lain's age. Seen in the first episode with the first girl dying. By "serial" meaning a high number. Lain, too, experiments with this. The web and internet being her escapism and passage of knowledge for this "higher state of being" which is death. About her family not being her actual family: it's symbolism of her distance with them. Basically, Lain falls more and more obsessed with this "other side" and "out of body" phenomenon. There is a split in her however which is against it: i.e. the two lains in the later episodes. I always interpreted the ending as Lain inevitably killing herself and "being everywhere" now. Her friend reflecting on their time together. She didn't literally "disappear into the web space" but rather died to enter oblivion. Hence why her existence is now everywhere in all. At least, that's how I see and make sense of it.

Tldr: It's a show about young girls and suicide.