Personally, it just seemed like a natural progression. I had been able to send message to people locally for quite some time via bulletin boards. Then things like fidonet came along where bulletin boards would connect to each other over a period of time and your message could get to somebody on another bulletin board. Email wasn't much of a stretch after that. None of it felt like "oh my gosh this is going to be huge!" It all felt like, "That makes sense."
Yep, I remember that too. It was a slow progression.
I first learned about terminals and modems because my dad had a VT220 for work. He would dial in using the 1200 baud modem. It wasn't "magic" or anything, it was just like learning how to make a phone call. Oh, you dial this number, you press this button on the modem when you hear this sound, and now the computer is connected and you can type stuff.
A few years later I discovered that the modem and terminal wasn't just dad's toy, but I could use them too, to dial into public BBSes. In those days BBS'es were standalone systems. You'd dial one up, you could post or read messages on its (sole) message board, you could play door games, you could upload or download files. You could call up with a dozen or so different BBS's one by one and start getting a feel for who hung out in each one, which one to dial into to talk about certain subjects or to download files of a certain genre.
The next step was FidoNet. BBS'es would call each other up at 3:00am and synchronize their message content. This was cool. Now I could dial into my favourite BBS, post a message, and within just a few days, that message would propagate to other BBS'es around the world (whoever subscribed to that particular node/network). And I would get answers back! From people in other cities! Cool.
By this time dad had purchased a 386 (25 MHz!) which ran DOS and Windows 3.1. Just in time for me to start junior high. (I had begged and begged, "Dad, can we please buy a PC? I know they're expensive, a 286 would be fine!" He kept on gruffly saying no all the way until he brought home the 386. I was so thrilled!)
I saved up my pennies and sent away by mail order for my own 2400 baud modem, which I was very proud to install and used extensively for BBSing. I even ran my own FidoNet "point" for a while, like a rudimentary dial-up email service.
Email technically existed at the time but most of us didn't use it yet; our concept of talking "privately" was more akin to private messages on discussion forums. It would be another few years before I found a large BBS that interfaced to "the internet" and gave me access to popular internet services like email, gopher, archie, telnet, finger. It had multiple phone lines, too, so multiple users could access it simultaneously, and, even cooler, actually talk to each other! You could send an "OLM" (online message) to a user on another phone line, and you could set a "who banner", a short message that appeared just below your username when someone typed "who", the command to show who all was online at any given time.
So, today, text messages and Twitter feeds and Facebook statuses, are light-years ahead of what we had in the 80's and 90's, yet in a way, not that different at all. It's just incrementally, evolutionary, better and better.