The phrase "to become one's own publisher" means to take on the role of publishing one's own work, rather than relying on a traditional publisher to do so. Full definition
That's because after more than 30 years of publishing highly regarded hardcover and paperback books — mostly serious nonfiction — I've become a publisher of essays for electronic readers. (publishingperspectives.com)
How does your product become the publisher for what your users do with it? (entrepreneur.com)
More scams are popping up, more agents become publishers, and so on. (deanwesleysmith.com)