No backdoors in your software.

The difference between "closed source" (proprietary) and "open source" software is (how did you guess?) that their "source" is open. Huh, okay, why do I care? Well, the "source", or "source code", is like the secret recipe of every software, like the recipe of a cake. When you buy a cake, there's no way you can figure out the exact recipe (although you can guess bits and pieces, "there's some coconut in here"). If a bakery gave out the recipe for its super-sucessful cheesecake, it would soon go out of business because people would bake it for themselves, at home, and stop buying it. Likewise, Microsoft does not give out the recipe, or "source code", of their software, like Windows, and rightly so because that's what they make their money from.

The problem is they can put whatever they want in their recipe, without us knowing. If they want to add a bit of code saying "every 12th of the month, if the computer is online, create a list of all the files that have been downloaded in this computer since last month, and send it back to Microsoft through the network". Microsoft probably doesn't do that, but how would you know, since everything is closed, invisible, secret?

A little while ago (October 2008) a lot of Chinese Windows users (most of them buy pirated copies of Windows) saw something strange happen with their computer: every hour, their screen would go black for a few seconds. Nothing to really prevent you from working, but it can easily make you go nuts. Microsoft had added a bit of code (an ingredient to the recipe) saying "if this is detected as a pirated copy of Windows, make the screen black for a few seconds, every hour". Now the point is not that the software was pirated: pirating software is bad, period. The point is that these users got an automatic update for Windows (updates usually fix bugs and add new features) without knowing how it would affect their system. No one knew.

Changing the source code of open source software is a much more open process. By definition, all the recipes are public. It doesn't matter to you since you won't be able to understand the code anyway, but people who understand it can read it, and speak out. And they often do. Every time someone wants to change the source code, all other developers are able to see the change ("hey man, why did you add this code spying on the user's keyboard input, are you out of your mind?"). And even if the whole team of maintainers for a piece of software go crazy and start adding puppy-killing features all over their source code, someone outside the team can very well take the code, remove all the bad bits, create a whole new version of it, and let the world know what the difference is. It's open.

That's why you can be sure open source software doesn't do bad things behind your back: the community keeps a close eye on all the recipes.