What's the incentive to answer questions?

I think it is like a game to many people.

The reasons… (in no particular order)

1. So they can climb up the user pages
2. So they can gain reputation to be able to have more ability in the system
3. So they can gain badges, just for the fun of having them
4. So they can help others because they like helping others
5. So they can help themselves; because, they often learn things by having people criticize their answers
6. Makes them feel good about themselves by helping others
7. Because they want to promote their own products or services in their 'about' description block.
8. Because they want to be known in the programming world
9. Because they want to expand their world of known topics
10. Because they want to get more in depth knowledge of topics they already know
11. They like to be heard, it's a good alternative to having a blog
12. It's fun to be higher in user rank than Jeff and Joel.
13. Because they want to be the one that dethrones Jon Skeet when he finally takes an extended vacation.
14. Because it improves your writing skills.
15. Because people like to be re-assured that they are right.
16. Because people like to know when they are wrong.

Engineers like to be right. They like people to recognize their skills. Stackoverflow is shrewdly designed to reward and reinforce these tendencies. That's why people answer questions here.

Because deep down we all want to fix problems, be it out friends's problems, spouse's problems, other random people's programming problems. We fix things, that is what we do. And really this is a positive outlet rather than meddeling where we are not wanted.

We also probably have at least a small bit of know-it-all in us, that is trying break free.

As a programmer, I always enjoy running into "atomic" problems that are mostly self-contained (like how to draw multi-colored text in a label or how to get live GPS data from a Garmin PDA), because I can focus on the problem without having my attention distracted by the bigger picture. In my day job, unfortunately, these encounters are actually kind of rare for me, because most of the technical challenges of my platform have been solved and because I usually turn these tasks over to junior developers.

StackOverflow is a never-ending stream of these atomic problems, kind of like Sudoku for programmers.

Most people just aren't utilitarian :)

What makes some kinds snap their fingers and shout "Oh, I know I know!" when the teacher asks a question? What makes some people angry at inefficient processes on illogical solutions even if they don't have to pay the price?

For me, that's the gist of it, even though there are some more possible explanations, that could be interpreted utilitarian:

It's leaving a mark in this place – some people plant a tree, some teach you how a garbunkulator works. It's seeing your name in print. It's gaining status in a community of peers.

It can also be a way to "validate one's profile":
I am a "development architect", supposed to assist 400 programmers in their development process. SO is a good way to test myself ;) And to learn lots of new topics.

I can only speak for myself and why I am replying.

My first reply was a quick and dirty answer that was (correctly) voted down. That made me put more effort into answering other question just to show that I can. And yes, the scoring system also help on the motivations to spend time on my answers.

Getting a positive vote also helps on the motivation. It's not the points itself, but the feeling that someone actually find my answer useful.

Another thing is that I am learning more from trying to answer. There have been a lot of questions I have tried to answer, spending time to search up more information, and ended up not posting any because someone else have given a much better answer before I was finished with mine. :-)

Because it beats answering questions on Yahoo Answers.

I like to answer questions in here and on irc programming channels, to improve my answers for questions by other people. By looking at what answers get accepted and how many reputation one gains, one can notice what answers were too advanced / too short / off-topic or something else.

Because we like to help each other.

For the same reason it's fun to click on monsters in Diablo. They've done a good job making the site fun.

I agreeing with the rep-collecting and badge-collecting answers.

Additionally, anyone who has benefited from a Question or Answer on the site most likely wants to keep it a useful resource. It's also a breath of fresh air and genuinely better experience that Experts Exchange and all the random sites you used to get when googling for an answer to your own question.

I'm embarrassed to admit that I also enjoy editing questions :P

I'm here because I'm trying to improve my English writting skills and learn more about ASP.NET MVC and when I'm doing nothing, I like to answer questions that I know.

Subconscious ego.

People like to share their opinions, and people like to be right.

There are lots of reasons, but my primary reason is to learn. Whenever I give an answer, I want it to be wrong so that someone else can correct me. To believe I have some knowledge and learn that it is wrong is an extremely valuable learning process which is best achieved by answering questions in a public forum. Also, I tend to give overly brief, incomplete answers, both an SO and in real life and SO gives me a chance to practice verbosity.

I often forget the solutions I come up with to the questions that I have. Then when I google the same questions a month later, it's really helpful to read the answers I've given to them.

One reason is altruism. Helping someone makes people feel good.

Another reason that is common on SO is to get rep because a big "impressive" profile looks good when you are hunting for jobs and a recruiter comes across your profile.