Three main things:
1. First mover advantage - There were others around the same time - Jaiku, Pownce, Plurk, Identi.ca, etc. - but they didn't get critical mass quickly enough.
2. "Disruptive" product - Who knew a new online platform could totally transform the way we communicate and consume content.
3. Open API to encourage ancillary app development. The rush for 3rd-party app integrations created a zeitgeist of innovation (and future acquisitions for the company).
I'd also say that the simplicity and focus on limited features was an advantage. And nimble response to its early adopter usage patterns and requests (ex: integrating "retweets" and hashtags into the product feature set).
(For context, I started an early Internet company in 1995, a social media marketing firm in 2005 before we called it social media and began using Twitter March 2007.)