First, a brief explanation of what Open Source Software is:

Open Source software is software that is given away freely. But more than that, the source code to the software is also given away. That means not only that anyone can use the software, but anyone can see exactly how it works. And even more, anyone can take that software, change it, and redistribute it. The quintessential example is the Linux operating system, but there is a very large collection of such software.

It also goes by the name Free Software. I actually prefer both the Free Software philosphy and name, but in this context they are fairly similar, and Open Source is used more often in a commercial environment.

Why Open Source Software is good for you

You don't read code, you don't write software, why do you care?

Well, the first and most immediate reason: it's free. Not only is this helpful because it costs less money, but it also means that there's a very low barrier to entry. If there's some particular piece of software that you think could be useful -- but you aren't really sure -- you don't have to spend anything to find out. And if you find you want to use twenty such pieces of software together -- all of them probably fairly small -- it won't break the bank to use them all. In fact, if you are using a Linux environment, you'll easily use a hundred little pieces of software: it's an environment that grew up with freedom, expects freedom, and makes good use of it. If each piece cost something -- $1000, $100, $10, even $5, the total system would become too costly. (Proprietary programs avoid this by bundling all the functionality in one large program, which leaves you highly dependent on a single software provider).

The second reason: anyone can use it. That means if someone else works on the code I wrote, they'll have everything I had. And, since I release my own software as Open Source, they'll have every right to change it and make it their own (though they have to let others do the same). This gives you more freedom. It puts consumers in control, even if you have to go through a programmer for many of the improvements you might want. It increases competition.

For someone else's take: The Open Source Initiative: The Open Source Case for Customers

Why Open Source is good for me

Because I can look at the source code for the software I use, I can reach a level of skill with these tools that would not otherwise be possible. Whenever I use Microsoft products, I find myself stymied by the opaque nature of the software. But even when Open Source software has warts, I'm able to understand those warts and even fix them if they are causing real problems. Open Source projects also tend to have much stronger communities. The entire premise is one based on sharing and creating for the community, so people are very helpful.

And, in turn, if I can work better, you get better results.

For someone else's take: The Open Source Initiative: The Open Source Case for Hackers (note that "hackers" is used to indicate a dedicated programmer, not an intruder)

Why Open Source is good for the world

Okay, you've gotten this far, now I'm going to show myself for the pinko I am. From each by ability, to each by need. This is what Free Software (and indirectly Open Source) is about.

With most belonging, for each thing I give to you, I have something less than before. I give you my food, and I will go hungry. With software this is not true. I can give you a copy of my software... I can give a thousand people a thousand copies of my software... they can each give a thousand people a thousand copies each... a million copies, and I have no less than I started with. There is no scarcity, except the scarcity of time to make new things. Why should we withhold what we have already made? Why should anyone lack that which would cost me nothing?

And do we not have a moral responsibility to share with our friends, neighbors, family? Who would I be to tell you that you must stifle your natural inclination to help? What kind of selfishness is it that I would not only keep for myself, but ask you to keep for yourself for my benefit?

Well, this high-minded thinking is what started Free Software. It's what offends some people, and why the watered-down philosophies of Open Source were offered. But this is part of why thousands of programmers produce software with little thought to personal gain, and this is why I use and produce it too.

Others attribute this to a gift economy: an economy where your status is derived not from what you have, but what you give away. There is a lot to be said for this. A good argument for this is presented in Homesteading The Noosphere.

For someone else's take on this philosophy: Why Software Should Not Have Owners (from the GNU Project/Free Software Foundation, who deserve credit for starting the movement, and carrying the torch all those years before it reached critical mass)