I've encountered an autistic man. How can I better understand the way his brain works?

Just choose to be a listener rather than a talker, do not try to get too close, give him space. Limit the time you spend with him and avoid judging in any way.

The best way is to ask him questions and listen. It will be trial an error. Just be patient with him.

http://themighty.com/2016/03/temple-grandin-explains-how-her-autistic-brain-works-like-a-search-engine/

I recently watched the film Snow Cake which features an autistic woman. I’m sure they are not all the same but it is a very good film and worth watching anyway.

There is this guy who wanders around town. He’s been here at least since I moved here in 1995. He’s known as “Sarg.” He’s probably my age and always dressed very neatly, in clean kakies with a very clean, maybe even pressed, button down shirt, tucked in. Nice, clean shoes. He sweeps things up around businesses. Sometimes he’ll go in and ask if he can wash windows. When I pass him on the street I always say, “Hi Sarg!” But I don’t make eye contact because it weirds him out. I once asked him if he needed a ride someplace and stammered, “No. No.” and he turned away abruptly. So I just nod and smile when I see him, and tell him the alley (or whatever he’s sweeping) looks great. Sometimes I get a glimmer of a smile and the very briefest of eye contact. I think he likes me.
I have no idea where he lives. I wish I knew his story.

It’s all trial and error because each case is different. The best suggestions above are to listen and be patient. It is a very time consuming exercise.

There is a book you might want to check out, called Son Rise. I first read it in the 80’s, and it was my first introduction to autism. It was the first time I’d ever heard of the word. After I read it I realized it described one of my daycare kids to a T. That book helped me approach the child and relate to him.
His parents said I was the only baby sitter they’d had that could deal with him, and he actually improved under my care. However, they refused to acknowledge that something was wrong. The father, especially, was adamant that the child was perfectly normal. — We never had a direct discussion about it, but over the course of 3 years I picked up on all the cues that told me they didn’t want to see it.
The book had an extremely unique approach (for the time) for dealing with, and understanding, an autistic child, and it pretty profoundly affected my outlook and my approach to everyone.

In a nut shell, the comparison is the autistic kid is is on one side of the bridge, you’re on the other, with this chasm between you. The prevailing idea at the time was to stand on your side of the bridge, and get the kid across that bridge by himself.
Well, this book says, “Hey, you go across the bridge to him, first, then lead him one baby step at a time to your side of the bridge…if he’s willing to come with you. Be prepared to leave him on his side of the bridge, or anywhere inbetween, if he’s more comfortable. Just go over and visit often.