Well I think that people who feel more strongly than "I don't agree with you", to the point where they are issuing death or rape threats, are deeply, deeply threatened either by what someone is saying, or by who that person is.
Lots of American racists are absolutely furious that Obama, a black man, has the temerity to be president. Not particularly because of his policies - though they don't like them, obviously - but just because he is a black man and as far as they are concerned, he has absolutely no bloody right to be in that whitehouse. As far as they are concerned, it's an aberration against nature - everyone knows that presidents are white men, and for a woman or a black man to come and turn that upside down, is deeply deeply threatening to them as it indicates a threat to their way of life; it is a signifier that the world has changed and they feel immensely threatened by it.
In the same way lots of misogynists simply can't bear to see a woman step out of line. And stepping out of line, is constituted by her saying anything they don't like, or seeming to be anything they don't like, or being in a place where she has no right to be. And in public, being listened to, is a place women have no right to be as far as they are concerned, unless she is saying something they want to hear.