As a writer who's made his living (more or less) via Medium over the past year, hopefully my experience can shed some insight.
The first blog post I wrote on Medium [0] got there by accident: I planned on publishing it on a popular local blog. While I'd used WordPress before, I didn't have a personal blog, so I wrote everything in Microsoft Word.
The piece contained a lot of images—so many that Word kept crashing. The deadline was approaching, so I decided to finish things in Medium and sent the link to my editor. (I probably could have used Google Docs, too.)
When it came time to publication—keep in mind, the editor and I hadn't signed any exclusivity agreements—the editor had not only published my piece with uninvited edits, but the image formating on their website was super unappealing. I decided that when promoting my piece, I'd link to my Medium version instead, since it looked better, didn't contain the edits, and the Medium.com URL carried no less weight than a URL from RandomMontrealPublication.com.
My hunch was correct. The Medium piece outperformed the piece on the "popular" Montreal blog I submitted to (it still did pretty well there, despite the editorial team not doing a great job of pushing it themselves). When local press began to pickup the story, all of the backlinks were going to the Medium version, rather than the RandomMTL version. One of the reasons it did so well on Medium was that it was selected as a Staff Pick (which put it in the newsfeeds for all users, as well as a newsletter sent to all users). For a first-time writer, the attention that Medium was able generate for me (a complete unknown!!) was super, super addicting.
Fast forward a month. My popular Medium piece lands me a job at a local startup as a copywriter, where among other things, I'm tasked with reviving the company's blog. Big surprise: the team has been swamped building their app, so the made-in-house blog is a bit dated. I convince them to try Medium, arguing that it has a native audience that can send us traffic.
While the "content marketing" I write for them never gets a Staff Pick endorsement (nor the corresponding Medium user traffic surge), it does get some pickup among the Medium community, including an Economist editor who emails us and references my piece in a story [1][2] and a Quartz editor who asks to syndicate it [3].
Eventually I get approached by a second startup (also in Montreal), who've seen my pieces on Medium. They have three writers on staff who have all published Medium #1 stories and we all continue to use Medium as a way to republish stuff from our blog (for traffic), and as an alternative to press release PDFs when pitching press (because they look good). Moreover, if there's anything we write that doesn't fit on our blog (like my last post, "The 50 Best HackerNews posts of all time" [4]), we can skip our blog and publish on Medium instead.
TL;DR: Writing on Medium gives you access to a big audience, which is great if you don't have a blog or if nobody reads your blog. Moreover, even if you have a vibrant blog, posting on Medium can give your pieces a second wind. It doesn't hurt that writing in Medium is a genuine pleasure. Compared to its competitors (Google Docs/Tumblr/Word/etc.) writing on Medium is like upgrading from a quill to a fountain pen.
While I've never managed to get another Medium Staff Pick, I always get a little traffic from Medium. My HN post from last week got 55k hits (many from HN) and Medium sent me ~2k of that. My latest post also talks a little about why readers trust the Medium.com URL, which may be another reason writers might gravitate to the site. [4]
[0] https://medium.com/@bagelboy/the-rise-and-fall-of-fairmountb...
[1] http://www.economist.com/news/middle-east-and-africa/2166421...
[2] https://medium.com/@transitapp/hello-nairobi-cc27bb5a73b7
[3] http://qz.com/357685/engineering-a-mass-transit-app-for-a-ci...
[4] https://medium.com/swlh/best-of-2015-pfffffffft-79d9b014f4de