You needed a computer, the right software (this was NOT trivial like it is today), and something called a GENLOC. The genloc mixed the video signal from your source and the subtitle signal from the computer. Then you recorded to a tape.
This master tape (or tapes) sometimes got out to distributers who did a lot of fansubs by owning tens of VHS decks. Some groups even had their own VHS shells made with their name on them. (Hence the faggotry began.)
Most of the time, you knew somebody who had tapes and you dubbed them yourself with two VHS decks. Or you could trust somebody and send them money and your address, and hope they send you tapes back. I got burned a few times that way. If you lived in a bigger area, anime stores or comic book stores would sometimes get tapes and rent them out, and you could copy them that way, too.