I've been wondering this, too, and design/ease-of-use aside, the biggest argument I've heard (and observed) is the network effect. Even trivial Medium posts get at least a handful of favorites and 1-2 comments. A built-in audience is likely an easy sell for the majority of folks.
Medium started out with Twitter sign in, which means their early users were all Twitter "influencers". The way I look at it, it's a blogging wrapper layer built around Twitter social graph. When you sign in with your Twitter account you automatically follow everyone you follow on Twitter on Medium.