I think the trolley problem is so interesting because it forces you to choose not really between whatever choices the dilemma offers, but between two methods of decision-making.
One way a lot of people make decisions is to use emotional methods. Those include empathy, attachment to other people, desire to be seen as a good person, desire to avoid the emotional consequences of making the "wrong choice" (either guilt, grief, or else fear that one will not be guilty over something that is indeed wrong);... lots of things. Emotional decision-making (I am not referring to "emotional reasoning" in the sense of making a choice to avoid insecurity, but in the sense of using emotions to aid decision-making) is actually a good thing, when it goes right. It's quick and reflexive and you often need to be quick in an emergency. It's what overrides the fear of whatever risk you yourself are taking. It's very right-brained. Emotional reasoning works fine for the vast majority of situations where the choice is simple--either interfere or don't interfere--rather than complex. It can break down and cause one to freeze in a complex situation, though, where both choices have a negative emotional value; and it's the reason we cheer fictional heroes who take a third option.
On the other hand, there's the logical part, the left-brained rationality that weighs the choices and picks the one with the best outcome. It can even be represented mathematically, if one is willing to assign values and probabilities to the possible outcomes. It's slower, but it's also more analytical; so it applies to all situations rather than just the ones for which emotional reasoning is valid. Logical reasoning can override empathy to let you take an active role that results in the deaths of some people and the salvation of more; but it can also cause a delay that results in the deaths of all --or even enough delay that emotional reasoning can no longer override the fear of risk on your own part. (There's no risk in the trolley scenario, but there often is in the real world, so I'm throwing that in.)
People who lose the capacity to feel emotions (usually through brain damage), oddly enough, do not become sociopaths. What does happen is that they become extremely indecisive. They are forced to depend entirely on logical reasoning, and this is a handicap because it slows down the reasoning process to the point that one might spend two hours deciding which socks to wear. That's because emotional reasoning lets you make a quick choice between actions of essentially the same value. Logical reasoning wants to choose the exact best option; and without emotions to make the "irrational" choice that you want white socks today rather than gray, you would be stuck on that choice. (I always want to advise doctors who treat patients with this sort of brain damage to try giving their patients some dice to make essentially-equal decisions for them; but most likely the patients would simply freeze deciding whether this decision had essentially-equal options or whether one choice was significantly better...)
Most people will, of course use some combination of logic and emotion to make decisions. I tend to use logic more (that makes me an atypical female, but a very typical engineering student); so I would be one of the people who, if forced to make a decision, decides to throw the switch, in effect killing a few people to let more live. That decision can only be taken so far, though, because "emotional" concepts like human rights, the value of human life, etc., come into it, too--so, for example, experimenting on prisoners (in such a way as to cause injury or death) to gain scientific knowledge that might save thousands of people is still unethical. My reasoning for this conclusion would be that the net loss from lowering the value of human life to allow such experiments would exceed the possible gain from the research because a lowered value of human life opens the door to other abuses of the same nature, which creates a society in which some people can be judged less worthwhile than others. (Another notable difference: Experimenting on prisoners is a choice that actually changes the way things are done, which is how it actively lowers the perception of human value.) That kind of society kills more people than would ever be saved.
The trolley problem has actually been used to survey people to try to determine which method people use to make decisions. It tends to be weighted toward emotional reasoning, but not by a whole lot. The most interesting gender difference isn't actually between who makes what choice, but how many people tend to try a third option. Men tend to reason in such a way as to assume they have already ruled out all third options; women will immediately gravitate towards third-choice possibilities. Maybe it has to do with the tendency to think in a linear versus non-linear way, since if you were a step-by-step thinker making that decision, you would naturally already have ruled out third choices in a linear fashion, while if you had a more associational style, you would be considering all the choices--including the third-choice options--simultaneously.