Why do Mormon missionaries travel in pairs?

There are some good answers here that use scriptural references. You may find this chunk from the church's missionary handbook helpful:

Preaching the gospel two by two is the pattern established by the Lord. The testimonies of two companions support each other in proclaiming the truth and bearing witness of it. Companions support each other in other phases of their work. They help each other learn and grow. They strengthen each other in times of difficulty. They can provide protection from physical danger, false charges, and temptation.

Two reasons that weren't mentioned in any of the other answers are: (1) to keep them out of trouble, and (2) to protect them from false charges.

(1) Most of the missionaries are young adults, many of them younger than 20. I hope this this doesn't hurt anyone's feelings, but 18-20 year olds tend to be kind of immature and good at getting in trouble. It could be something as trivial as breaking into an out-of-use building and getting arrested. Things like this could be bad for the reputation of missionaries in general, give the mission president a headache, worry their parents back home, etc. In situations where missionaries are in a foreign country on a visa, things could be even worse. If there is another person with you 24/7, these kinds of things are less likely to happen.

There are also more serious concerns, and though the church tries to only allow people living in harmony with the church's beliefs to go on missions, sometimes people get through the application process who shouldn't have. Because you can't be 100% sure any given missionary doesn't have serious issues, even if you think you know them, it prevents some potential problems if they are in pairs.

(2) It is a general rule in business that you avoid being alone with clients to protect yourself from false charges, and because missionaries meet a lot of new people and meet with people in private settings (peoples' homes), they also adhere to this rule.

The following excerpt from the church's missionary handbook discusses missionary rules about staying together (when it says "companion," that means the missionary that you are working with):

Stay Together. Never be alone. It is extremely important that you stay with your companion at all times. Staying together means staying within sight and hearing of each other. The only times you you should be separated from your assigned companion are when you are in an interview with the mission president, on a companion exchange, or in the bathroom. (For guidelines on baptismal interviews, see Preach My Gospel, p. 205.)

Never make exceptions to this standard for activities that seem innocent but take you away from each other, including being in different rooms in the same building or in a home. Situations that seem harmless at the beginning can quickly lead to serious problems.

If you live in an apartment with more than one room, always sleep in the same room as your companion, but not in the same bed. Arise and retire at the same time as your companion. Do not stay up late or get up early to be alone.

The quick answer is that they do this for the same reasons other Christians engage in missions the same way. Paul talks about this in 1 Cor 9:

Don’t we have the right to bring a Christian wife with us as the other apostles and the Lord’s brothers do, and as Peter does? 6 Or is it only Barnabas and I who have to work to support ourselves? ... Don’t you realize that in a race everyone runs, but only one person gets the prize? So run to win! 25 All athletes are disciplined in their training. They do it to win a prize that will fade away, but we do it for an eternal prize. 26 So I run with purpose in every step. I am not just shadowboxing. 27 I discipline my body like an athlete, training it to do what it should. Otherwise, I fear that after preaching to others I myself might be disqualified.

It all boils down to:

1. Accountability
2. Distraction
3. Pure Witness

At its most basic, evangelizing in pairs builds in accountability, a mutual commitment to a shared goal, reinforced by the social pressure of knowing another human being is aware of whether or not you are adhering to your commitment. Additionally, there is a certain encouragement. As Ecclesiastes 4 says:

Two are better than one, because they have a good reward for their toil. 10 For if they fall, one will lift up his fellow. But woe to him who is alone when he falls and has not another to lift him up! 11 Again, if two lie together, they keep warm, but how can one keep warm alone? 12 And though a man might prevail against one who is alone, two will withstand him—a threefold cord is not quickly broken.

Even outside of LDS missions, just about every reputable missions agency has some form of accountability, in order to ensure that the missionary is adhering to his or her commitment.

Likewise, Mormon men and women do not go on mission together for the same reason that Baptist, Catholic, or other missions agencies tend to discourage opposite-sex missionary teams - namely doing so detracts from the mission. When, for example, the woman who would become Mrs. Affable Geek was a missionary in Nepal, she had signed a document stating that she would not date on the field. (True story: I never dated my wife till after we were married as a result, although we did "fellowship" alot!) The author of Hebrews alludes to this saying:

... let us also lay aside every weight, and sin which clings so closely, and let us run with endurance the race that is set before us

The distraction of trying to build relationship (not to mention the perils of losing your witness i.e. causing others to think that you were living in sin / having sex, even if you weren't!) is not conducive to the purpose for which you were sent. A husband and wife pair is different - they've already gotten through that.

Paul modelled this in his own journey -

1. He travelled with Barnabas, both for encouragement, and so as to have a witness that he really was now with the church.
2. He openly talked about how having a wife would distract him from the mission.

As such, the LDS practice is both practical and Scriptural.