In fiction, I often have other written work for that author which makes it pretty easy to emulate the style. And in non-fiction, we have extensive interview time where they're telling me the story from their point of view. I get little colloquialisms, their personality, how they felt about the event from how they speak. It takes a fair amount of analysis and observation, but I record all my interviews and transcribe them and read them multiple times.

Then, of course, my client goes over everything I write. I tell them if they wouldn't say something, or they feel they'd say something slightly differently, to let me know. They'll usually go through and oust anything that doesn't feel like it fits.