Okay, so here's how it goes:
http://www.cartoonbrew.com/ideas-com...riven-era.html
^That article pissed off a few people in /co/ (4chan's cartoon board). Because MLP was mentioned in it quite a bit, quite a number of people went to go watch the show. It just so happens that the creator of MLP is also the creator of Powerpuff Girls and Foster's Home (the former was something I watched occasionally 15 years ago or so and Foster's Home for Imaginary Friends was pretty awesome).
The rest of /b/ found out eventually and by the second episode, many were hooked for different reasons, whether it be themes and characters or the bright artwork in contrast to many cartoons produced in Western media.
And being that it originated on /b/, the source of many memes and images (9gaggers, disagree all you want, but it's true), it spread rapidly, combined with the fact that Hasbro didn't really take MLP episodes which were being streamed on Youtube, down.
Eventually, there was a huge shitfest in /b/ and MLP was the center of the argument, between users who loved the show and those who didn't.
It also helps that a lot of the developers for the show have made accounts on Youtube AND 4chan/reddit (yes, including Lauren Faust; she posts on /co/) to interact with the demographics and receive feedback.
Celebrities and famous figures have also referenced the show, and that goes a long way in advertising something. Bill Clinton was asked about characters on MLP on NPR and Colbert gives a shoutout to "bronies" on a particular episode.
Long story short: Started out as a movement to prove an article wrong, then exploded into an internet trend.