In my case, it was a desire on my husbands and my part to learn and grow sexually. We both had no previous sexual experiences before each other so we thought it would be fun/interesting to 'date' other people. We established boundaries and expectations and then just went to dating sites to find people we wanted to get to know.
We weren't familiar with 'poly' at the time, thought maybe 'swinging' was what we wanted to try, but neither one of us was comfortable with the idea of that. I easily found men to chat with and met 2 in person but there was no attraction or connection. Those men wanted casual sex but it didn't appeal to me. Then I met my boyfriend *sigh*
I wasn't expecting to get so involved with him, but when my husband and I realized that I had I found a real connection with someone else, we soon learned what polyamory was and the rest is history! It was much more difficult for my husband to find a woman to date and he got very discouraged for the first few months. But then he found his girlfriend
From what I've read on the forums, there's no specific way to begin a poly journey, as with all things poly, the possibilities are endless! Some people have someone in mind, some people have affairs then are able to establish a more open and honest poly relationship out of the aftermath. And some people talk about the idea of poly and pursue it 'from scratch'. I don't think I would recommend the 'affair' route, but being able to talk about it beforehand is a good foundation to start