I'm going to preface this post with the fact that I have not read the books. However, I'm also not going to speak to the content of the books, but rather of the debate surrounding them.
I suspect the furor over the books stems from several different issues that people are having.
I suspect the most dominant cause of rage against Twilight stems from the inherent insecurities that people, and specifically young men, have. The societal issues in the United States surrounding young women is well documented; the same issues applied to young men less so. It's my belief that all people suffer from insecurities, especially during their more formative years. Leaving aside all other issues, we have a story that has captivated many women and provided an outlet for fantasy. The males are portrayed as, essentially, supermen who are worshiped by women in fiction and in fact. That has triggered a fair degree of insecurity among men, as they are compared to these supermen and in their minds found lacking.
In fact, this is the male version of one of the biggest problems facing women in the United States* society today.
It is also, of course, a simplification of the issue. It ignores the issues surrounding the female image of a perfect male. It ignores the cultural restrictions on the expression of male insecurity. It ignores the question of physical versus emotional versus mental maturity (ie, older women obsessing over Edward).
A second issue is the sensitivity of the issue of abuse, physical or otherwise. As has been mentioned above, it is a very big problem here in the US. I would bet that everyone in the country either has been the victim of such behavior, or knows someone who has. Whether the books do or do not discuss such a topic I don't know; I suspect that with such mass appeal, they likely do not. Regardless of the author's intention, that will be something that will be discussed, and should be. Just because a book may not have been written with inherent meaning** does not make it immune to literary analysis and criticism. That the series is so popular only makes it more of a target for these analyses; just look at any popular fiction over the decades, from The Odyssey to Lord of the Rings and Harry Potter. Given the subject matter of the book - the trials and tribulations of love - an analysis is bound to look at such things as victimization and abuse, especially with scenes in the book as described in this thread***.
This is amplified by the intensely personal feelings involved in the matter. Perhaps it was an ex that was abusive. Perhaps it was a crush that dated a guy that abused her emotionally. Perhaps it was a friend, or a relative, who was stalked and harassed by someone. There is no possible way to separate the subjective viewpoint from the debate, nor should it be removed. However, with something so emotionally charged, there is going to be more heated response.
A third, less subtle issue, seems to me to be the issue of American culture on the self-esteem of young women. It is no secret that our culture is especially damaging to them, nor that there is a large counter culture**** that believes young women should be secure in who they are, and not the unrealistic superwomen^ of our culture. Anything that is fashionable amongst young women, whether deliberate or not, is subject to scrutiny with those criteria in mind. I suspect this fuels a part of the debate as well.
* I won't speak for other cultures, since I do not know them as well.
** I don't know if it has, and this seems to be a large point of contention regarding the analysis of the book, specifically regarding the Mormon undertones present in the work.
*** For better or worse, either way.
**** I deliberately avoid using the term "feminism" here, as I believe it carries several negative connotations due to the actions of some, but not all self-described feminists. I've found mentioning feminism is a sure-fire way to get some people to roll their eyes and stop listening. Nonetheless, I suppose it would be the proper term.
^ Much as in the first paragraph. Also, I got sick of typing asterisks.
