I think it's a mixture of first mover advantage (it was, I believe, first out of the gate for online ticket sales), excellent marketing and, recently, a very slick app (no TOC comes close) along with a slick website. Giving it a fantastic advantage when it comes to name recognition and a solid points for usability. Couple that with people generally expecting to pay booking fees (they crop up in almost every other walk of life) the disincentive of the booking few is significantly reduced. Especially on high value ticket sales. If I'm spending the thick end of £100 on a ticket for long distance journey do I care that much about £1 or £2 booking fee? Well, I do to be fair ! But I'm sure for many they don't. They also have the advantage of the fragmentation of the industry for instance I wonder how many people think you can only buy tickets for TOC A from their website and you can't buy tickets for TOC B but if you use the Trainline they can sell you tickets for anyone! As well as the catastrophic state of ticketing in general feeding into a sense of consumer confusion which I'm sure helps the Trainline with their slick marketing to emphasise the 'benefit' of their service.
That being said I am a Trainline customer! I use their app to buy tickets for on the day travel for which they charge no booking fee as I still find their app to be by far and away the best designed of anyone from TOCs or third-parties for ticket retailing and presentation of e-tickets.