Universities and government institutions would lease lines from the phone company and use them to connect their computers to each other. The "network" and the "computer" were basically the same thing. My university's big computer connects to a few other big universities (and so on, and so on), and those computers would pass messages back and forth. These connections were like long extension cords- 1:1 connections. My computer (and budget) has room for 10 connections, and I connect to the 10 most useful computers.

Eventually this network of computers got too big to manage as a bunch of peers connecting to a bunch of other peers. Each site would need to connect directly to any other site they wanted to deal with regularly, or suffer really slow service. Further, they were spending more and more of their processing time passing other people's messages instead of doing work. The phone company said "instead of leasing theose 10 lines connecting to 10 different other sites, why don't you lease this one line and we will deal with delivering the messages where they go. If you connect to us, you can connect to ANY of the sites we have on our network."

The phone companies were the first ISPs.

(This is the difference between circuit-switching and packet-switching, for further reference.)

Too young to remember dial-up modems, I take it?

They connected through the phone lines, just like everybody did up until cable internet became a thing. You plugged a modem into your phone jack, and the computer would dial the phone number of the server that it was trying to reach. It would go through just like a phone call (including having to pay per minute for the connection), with the computer on the other end also having a modem plugged into their phone jack.

This meant, of course, that you couldn't use the phone and the internet at the same time (unless you had two phone lines). You also didn't have "the internet" in the sense that we have it today at first; what you had was a lot of independent computers/networks that you could dial into. "The internet" at its inception was more along the lines of a set of protocols (such as http or telnet) that made those connections possible. Connecting to some particular site or service required that you had their phone number.