There is a button on this form "Answer Your Question" How is this used? and What is its intent?
Is it an option when I come-up with a solution to my own question that I post it?
Yes, it allows you to post an answer to your own question.
When posting a question, you can click the "Answer your own question" check box, if you know the answer to the question you're posting.
Once posted, you see the "Answer Your Question" button, which allows you to post an answer to your own question after it's posted.
Note that you cannot accept your own answer to a question until the question's been up for at least 48 hours, and that you get no reputation points for the accepted answer.
See "Can I answer my own question? " in the help center for full, current details.
The intent of this is so that if you have a question that you can't find the answer to on the network, but then work out the answer, you can be a good citizen and post the question and answer at the same time to help those in the future that might run into the same problem as you.
An example of this is a question I posted on movies.se. I was watching a show, had a question, and then figured it out after some thinking. I looked and the answer wasn't on the network anywhere, so I posted it so that future searchers can find it and get an answer.