There's a paradox that if you're telling the truth in the interview, there should be no reason for someone to feel the need to question it. The story will hang together, nothing will stick out as odd, and you can get on with talking about the tech you used, the skills you have, and the fit for the new position.
If someone needs "proof" of something that you know is true, you don't need to work for them and it's likely some kind of "dance for my entertainment" rather than a genuine "I'll hire MB, but only if he can prove this one thing with paperwork."
Sometimes people won't believe you, even with the facts in front of them, because their world is too small for them to conceive that what you did is possible. You don't want to work with those people, but they exist.
I had a fairly senior hiring manager for an investment bank call me a liar to my face, basically because he couldn't understand how a consultant can have more than one client at once, and so my CV is non-linear! I dodged a bullet there.
I had a similar experience, 20 years ago. I was one of the first publishers to be accepted into Google AdSense in my country. I run a forum and had my blog on the same domain.
Initially Google used to send US cheques, in USD. So I had to go to the bank to deposit those in my personal account, which required a branch manager to sign it because 1. Who is Google and 2. Why is someone getting a large-ish cheque from a company overseas (AML laws).
Anyway, at some point I've decided to incorporate a company. Trying to explain to my accountant what my company was doing (online publishing), and trying to explain to the bank this was not a money laundering operation required some effort.
People had no idea that one could make money on the Internet.
Advertising for them had to fit in a TV, billboard, radio or cinema square. Anything else was science fiction.
I then left my 17-year full-time as a mainframe developer (yep) to run my company full-time. You can't imagine the run around I had when I tried to find a home loan for our first home a year later - how do you make money? was always the question. It didn't matter that my answer was "I am the owner/director of a company". They wanted to know what my company was doing and just "online publishing" wasn't enough. Even with tax records and everything.
So I empathise with the OP on how to demonstrate skills and experience while being self-employed.
Oh yes, the whole "but you don't have a regular salary and spend it all every month" thing was a problem getting a mortgage for me also. Even though I got a pretty darn good consulting rate in banking and paid the whole mortgage off in very few years in the end... No one could officially understand "I only take out money when I need to" but happily I had a sensible branch manager in the building society that I used...
If someone needs "proof" of something that you know is true, you don't need to work for them and it's likely some kind of "dance for my entertainment" rather than a genuine "I'll hire MB, but only if he can prove this one thing with paperwork."