There are a lot of factors that play into this issue, not all of which I feel qualified to fully discuss, so I’ll link where I can to more knowledgeable sources.
Personally, I think a lot of it is just misunderstandings and different ways of viewing the world, what definitions of gender, etc. really mean. Here is a good discussion of this from a Mormon perspective. I also think on some level it comes down to the fact that the Church has outwardly stated it supports traditional ideas of gender/family (there's a famous thing called The Proclamation on the Family-- I understand if you don't agree with it, but it has become official doctrine of the Church and should help you understand a little better why this is an issue for us) and feels like it should back that up when it's directly challenged, as with Prop 8. I've heard so many arguments on both sides and I don't like how they're playing out on either side (the anti-Mormon talk or the anti-gay sentiment some people strike up in order to explain to themselves why the church would speak directly against this), that I'm quite torn, precisely because this is such a hard issue.
Personally, I support giving full partnership rights for everyone who wants them, but marriage I think is a religious institution and should be defined individually by each religion. I think we should adopt the European model, where everyone goes and gets a civil union from the state and then goes and does whatever religious marriage they might want, and then individual churches can decide to perform or not perform gay marriages. Those who still want “marriage” without the religious context could maybe have separate ceremonies of their own, but marriage would be individually or institutionally based and recognized. But we're a long ways from that. One reason I was a little against Prop 8, though, is because I think any state definition of marriage, whether for or against gay marriage, is abuseable and dangerous. But I think I'm a little in the minority there.
One small clarification which I'm not sure is true but might clarify why so many Mormons were scared: on the objection that many had that the issue doesn’t directly affect Mormons, there was a lot of buzz going around that if gay marriage were legally sanctioned then it would be legally required to accept/perform gay marriages and institutions which refused to perform such would no longer be licensed for marriage, i.e. the Church might lose the ability to performs marriages and they might even shut down our temples in California, which is scary beyond all reason to Mormons. That immediately reminds us of burning temples, being driven from homes/cities/states, massacres, and other really nasty things that happened to us during the early days of the Church, when it was made legal by the "Extermination Order" from Governor Boggs to kill Mormons in the state of Missouri, etc. (Funny thing, the Extermination Order was only officially repealed in 1976.)
The really ironic part of this whole thing to me is that the gay and lesbian community thinks the Church is trying to stamp out their rights but on some level the Church is afraid of the exact reverse: losing their rights again. It's two completely opposite but similarly marginalized groups fighting each other instead of getting mad at the greater society.
I'm not saying this paranoia is warranted (because I really don't know if they could shut the Church down like that, though the idea scares even me a bit) but it's one of the reasons why Mormons really went in on this one.
It’s obvious there are a lot of views that differ about the nature of what gender, orientation and relationship mean, and how we should react to those on a societal level. I consider myself a feminist because I do believe in gender equality and in the freedom of choice, but I definitely think anywhere where the law gets involved is messy, because the law is there to limit freedom for the greater good (we don’t have the freedom to kill people so that others don’t have to worry about being killed). How that greater good is defined is up to the majority. So freedom of choice is a complicated issue, as is gender equality. On some level I do believe that men and women are inherently different, but I believe that makes both more valuable rather than less, because both are unique. I don’t understand everything, or even very much, about to what degree we are the same and to what degree different, but learning more is one of the main quests of my life. Mostly, with this post, I hope you see that Mormons are 1) not all of the same mindset, 2) still people, even if you very heavily disagree with them, 3) trying to do their best to stand up for things that are important to them, even if they might do it in ways that do more harm than good.