Medium launched in 2012. It's almost 6 years old which is not "rapid rise".

They started with a very good editor and a syndication model that is inherently viral (i.e. every post on medium promotes medium itself and other posts on medium). This is much better than most other blog hosts like WordPress.

Tumblr is similar and also very popular but Medium focused on high-quality content and not short posts like Tumblr. They even were publishing their own content for a while.

They also tried a hundred different things, many of which don't exist anymore.

In short: good product, built-in virality and lots of grit and experimentation with different features.