There are radical feminists. They have always existed within feminism, just as radical <whatever>s exist within any political group of <whatever>s. I don't always agree with everything they say, but I believe their existence is important vis a vis advancing feminism as a whole. By pushing radical ideas within feminist circles, radicals are constantly dragging feminism away from the status quo, always putting it in conflict with the status quo, which is pretty much exactly how social movements are supposed to work. (As an example, see the evolution of Catholic and Christian faith over the course of modern history.)

We are probably living in what will one day be called the Fourth Wave of Feminism. The first wave was focused on women's suffrage. The second wave was focused on women's right to work (and to earn money). The third wave was a more reactionary response to the second wave, and as such didn't really have any concerted objective though it carried on many of the same issues that concerned later second wave feminists, such as reproductive rights and sexual violence. In particular, what happened was that after the second wave, the feminist movement basically imploded into infighting groups, and so third wave feminists mostly fought their own battles in small groups, and were often in conflict with one another on what feminism was "supposed" to be.

I think that we're coming out of the third wave now, and one side has basically won. Notably, in this case, it seems to have been relatively radical groups that focus on racial, gender, and sexual equity together, which I will casually combine together under the umbrella term "intersectionality". I'm not particularly certain as to why this faction has "won", though it may be tied to the Internet providing increasing access to women who were previously marginalized (e.g. poorer, uneducated, working-class women), increasing levels of immigration in the past resulting in young, well-educated, but also racialized women, increasing levels of globalization, and the decisive cultural victory by gay/lesbian activists in the American culture wars of the 90s. Notably, many older second-wave feminists are being repudiated for being too "traditional", as are many theories that would probably have seen great interest during that time (e.g. Lean In). We also see many women, particularly those who are white and wealthy, repudiating feminism back, when they would have previously been the key players/role models/target demos for feminism.

So basically, at this time, feminism is basically escaping the box that it's been lying comfortably in for the past twenty years or so, and it's doing so in a very aggressive, radical way. It's explicitly trying to draw strength from other social justice movements, past and present, and linking them together. It's powered by a generation of young female millennials who don't particularly seem to care for the term "feminism" and its strict etymological meaning, and are rather willing to take on all comers (rhetorically). This is, I think, where the "SJW" term really originates: they're no longer restricting themselves to just issues concerning (all) women, but across the whole spectrum of social justice.

The target of their ire is systemic inequity, which is manifests through resistant institutional bureaucracies and societal structures, and through cultural pressures. This, again, extends from second-wave feminism - its failures, principally. As a movement that tried to work with the system from the top-down, pressuring politicians to enact laws to protect women's rights, its failures are now considered the result of a wrong-headed approach. The correction, then, is to work from the bottom-up, to destabilize the system rather than cooperate with it. The objective now is not to get the government to say, "Do the right thing," though certainly that remains key in certain areas (such as reproductive rights), but rather to force individual organizations and people to "do the right thing" on a case-by-case basis. They are also, I think, quite intent on winning a cultural conflict, possibly drawing from the strategies and experience of gay/lesbian activists, but also quite sincerely because they believe that changing society's tolerance for anti-women culture is part of the path to equity.

Given all this, the popularization of the term "SJW" as a pejorative is then pretty predictable. Evey major breakout of feminism (or any social justice movement, really) creates a natural backlash among those who are ideologically opposed, for whatever reason. When women were marching in the streets for the right to vote, they were targeted by bombs, beatings, and worse. We add to that the fact that they're taking on all manner of social justice issue, and so, just as their coalition broadens by incorporating in activists concerned primarily with other social justice issues, the coalition that opposes them also grows. Adding in the fact that each unjust and/or hostile action is viewed as contributory to a larger, pervasive anti-woman culture, plus a society that hasn't quite figured out how to work social media and what our expectations of it are, means that this conflict has many, innumerable flashpoints, which keeps this conflict burning very hot and bright.

This has created a polarization in the realm of social justice ideology, with people arrayed on one side or the other (and also the many who are just sitting it out, or working quietly on more "moderate" proposals absent the labels of either side), and while there are differences of opinion between those on the same side, now everybody gets tainted by everybody else's actions. So if you advance what may objectively be a reasonable position on women's rights, there's an assumption that you're an SJW who's on "that side". Or if you're actually concerned about ethics in games journalism, now you're also one of those Gamergate stalker/harassers out to destroy individual women's lives. And, of course, since they're taking all issues, they're not just "feminists" now, but "SJWs", which then naturally positions those who in the opposite as opposing social justice. In truth, neither side really has a truly comprehensive vision of what social justice entails, and both sides are fighting for what they should be considering social justice. It seems incredible to me that anybody would be seriously saying, "No, we're for social injustice!"

Anyways, that's where you are: in the middle of a major ideological battle between two groups of people both of which I'd largely consider radical. If things don't make any sense to you from the outside, that's because everybody's coming to this battle with a lot of historical baggage and convoluted backstories because of how this particular battle has shaped itself.

Fake-edit: Oh hey, there's actually a Wikipedia article on Fourth Wave Feminism now. I totally didn't know that before I wrote this.