Why should I write a good undergraduate thesis?
The audience
The correct, but somewhat cynical answer is that the main audience for a thesis or dissertation is the examiners. They'll have to grade it (or at least sign off on it), so you want to convince them to give you the grade you want. The next audience you'll want to target is actually yourself.
Some years from now you'll have forgotten some of the details. You can try to track them down again, but if you've written stuff down in a format that's helpful to you it's going to be a lot quicker and less painful. Basically, if you write the document well it can be a great reference. (I admittedly haven't opened my bachelor thesis many times, but I looked something up in my Masters thesis just yesterday.) Sometimes the thesis can also be useful for students doing their projects on the same topic, in the same research group, etc.
The other big point is that writing the thesis is supposed to be practice for future writing of e.g. dissertations or papers.
What sections are most important?
From the practice angle, all of them. To convince your examiners, probably introduction, results and conclusions - but different people will have different priorities. However, it's a case where you have a few people who are supposed to read your thesis, so an excellent abstract is less important than in a research paper. Basically, you want to show that you understand the field, and have done good work.