By falling out with an existing preserved railway, thus setting up another outfit in the same geographical area?
A number took over existing, operating railway lines - industrial, mineral or mainline. Others started as a campaign to save a threatened service (Beeching days or more recently) which, following closure of the line, then turned their efforts to re-open the line - some within a few years and before the track had been lifted, others many years after closure and with complete re-instatement required.
Nowadays, I suspect it would be a lot more difficult than in the 1960's - all the safety cases etc. requiring full-time professional staff and expensive outside advice from the outset. Also, in the past, there was a large pool of railway staff / recently ex-railway staff interested in setting up such ventures because they loved the job so brought their knowledge and experience with them.
I don't know of any statistics / research on the matter but there seems to remain a lot of links between preserved railways and the big railway (in terms of NR / TOC staff also volunteering at preserved lines).