The show was written and developed by Lauren Faust, who was also behind Powerpuff Girls and Foster's Home for Imaginary Friends, among others. The characters are all vastly unique, and the situations and humor can be appreciated by anybody regardless of age.
Naturally, the show's demographic was for young girls, and as such, each episode ends with a letter to Princess Celestia explaining what Twilight Sparkle learned about friendship during the episode.
From what I've gathered, the show's hype originated on 4chan's /co/ board (Comics and Cartoons), and they were just way too cute to ignore. So people brought them onto the /b/ board, where people took the facial expressions and interesting movements/interactions, like Dasva mentions, between the characters and made them into absolute perfect memes (like 'umad?', 'haters gonna hate', etc).
Rainbow Dash even has her own memes! '10 seconds flat' and '20% cooler'.