I've always felt that one of the show's greatest strengths, at least in appealing to me, is that, contrary to some of the above posts, it challenges the stereotypes of what "girly" means, rather than enforce traditional gender roles. Make no mistake, the show is definitely created with little girls in mind. Slumber parties, dresses, make-overs and adorable things all rear their frilly, perfumed heads with more frequency than cartoons aimed at little boys or at no gender in particular. But then, I've yet to see a girl's cartoon present the protagonists with real danger, such as dragons or hydras, and even physical violence. So if girliness comprises such things as well, then I suppose the show is girly. I suppose you could even say the mane characters act girly, in that they have as varied and opposing interests and personality traits as real girls and women. But that's where the show actually challenges what we think of when we think of girly things.
Case in point, Rainbow Dash is at first blush the athlete archetype. The G3 equivalent would be Scootaloo. But the similarity really ends at them being athletes. G3 Scoot is only an athlete inasmuchas she is seen doing active things more often than the other ponies. Otherwise, she has the exact same relationship with each pony that each other pony has with each other pony. That is to say, the relationship between any two ponies could be lined up with the relationship between any other two ponies, and they would match completely. Moreover, G3 Scootaloo's proclivity for sports doesn't preclude her from participating in even the girliest of activities. Contrast with Rainbow Dash, who first off has totally unique relationships to every pony, which are not repeated between any other two ponies. She also has very little patience for any activity that does not relate to physical activity or competition. So slumber parties and dress-up are out.
I could go on with every member of the mane cast, but that would get tiresome. What it comes down to is that the show isn't just some vapid sideshow of how many times the same character can be recolored and given a shallow "interest." Most shows that are of this type are considered "girly." And that's the real problem there. "Girly" automatically has negative connotations because of those shows. The fact that each character in My Little Pony Friendship is Magic is their own person, while still being girl's show protagonists makes the show singular among girl's shows, because being girly is now pretty darn cool, especially when it means you can kick a dragon in the face and still be seen as one of the girls. I consider myself to be a feminist, and so a show that says this appeals to me on an ideological level.
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But not everyone that enjoys the show is drawn in by the whole girl power appeal of the show. And let's not forget that a huge, overwhelming majority of the online viewership is male, so it's not like this audience is particularly empowered by the show's message. I do think, however, that the ultimate appeal of the show is rooted in a similar concept. But coming to that point in enjoying the show is a process.
First off, a very, very small minority of the show's adult male audience started out sold on the whole idea of actually tuning in every Friday to watch a show made for little girls, or generating actual original content based on the show. Most of them started out with some degree of skepticism. With most people who watch the show, something blows that skepticism out of the water. What does this is usually one of three things, or maybe two or all three. For one thing, the show is astonishingly well-animated, especially considering that flash (which is what all of the show's animation is done in) is usually associated with lazily produced, low-quality mini-games and animated shorts. And moreover, it's a welcome break from the animation in most other shows out there today, especially the crappy 3D model animated trash that's been showing up with troubling frequency. Second, the aforementioned character diversity. Not only is this appealing, from a literary standpoint, to the more educated adult audience, but it also ensures that one character or another will endear themself to a new viewer almost immediately, which in some cases is enough to get some people hooked for life. The final reason is nostalgia. The show downright lifts whole sequences from the Looney Toons, and in general feels like old Nicktoons, back before the only good Nicktoon was Spongebob. For some, the show just makes them feel like a kid again.
Some people may have their skepticism defeated by one of these qualities of the show, but it may not be enough for them to get over the sense that, somewhere deep down, they feel self-conscious about watching a show for little girls. But what seems to win these people over is the fact that there is a massive community of people who are reassuring you that just because you watch a show for little girls, it doesn't make you less of a man. While this community extols the show's ability to bestow lessons in an entertaining way, one of the biggest lessons the community seems to have taken from the show is: if "girly" can mean whatever a girl wants it to mean, then why can't I decide what "manly" means for myself. Indeed, one catch-phrase of bronies is, "MLPFiM is a manly show for manly men," a claim that would be absurd if one were letting cultural norms dictate what it means to be a man. The position appeals to those adult men who chaffed under the imposition of what it meant to be male in America. They weren't athletic, or they liked fashion, or they liked adorable things. Even though these young men were straight, they may have at one point been called gay or *** simply because of their inclinations. The fact that the MLPFiM community is saying that you can be manly even if you like a show for little girls reassures the doubtful, and speaks to those men who had never felt affirmed in their own masculinity because of the things they were interested in.
Ultimately, the show's empowering stance seems to have mobilized a generation of young men to empower themselves to challenge the gender norms they themselves never fully embraced. And now that the internet has given them a way to network, they've found that they're not alone. Essentially, the community created a chain reaction. Without such a community, I think the whole thing would have died down, but because of the community, the popularity of the show has grown exponentially.