Two things:
1. With computers becoming more widespread and with the advent of consoles the video game market greatly expanded, games were no longer the hobby of nerdy men;
2. And more importantly, with games becoming more popular, more women started playing them. Including RPGs. The end.
Romances are not put in there just to please women or non-nerdy men. They are created just as much - if not more - exactly for all the basement-dwelling nerds who haven't lost their virginity yet and use them as a replacement for the real relationships they can't have.
When did romances in RPGs start becoming popular? Let's take Bioware, around DAO or ME2? Before that romances were pretty inconspicuous, relatively speaking. Little more than just another role-playing device.
Mechanically, I'd argue it's when Bioware switched to their 'hub-style' gameplay where all (or most) NPC interaction happens at a campsite/ship/whatever, and the player is responsible for going from companion to companion to chat them up. In BG2, it was all scripting driven which (when it worked... because there were tons of bugs) ended up making it seem less intrusive and more part of the basic game.So... that would be KOTOR/Jade Empire?
If you mean romances becoming the point of gameplay for some people, I'd guess ME... once they added 'decent' graphics and sex scenes (such as they were), the general public started paying attention. And once that happened, there was suddenly a need for everyone to have a romance that catered to their demographic, and once that happened, there was an oversized amount of effort, writing, and dev time going to romance development.
It's hard to pick a specific time though, given that there were societal shifts (and even the start of changing demographics in dev teams) occurring simultaneously.