So I send out the final publicity letter for the contest, and I get a question.
Well, that not really a question, as it is an implied question. Nevertheless the message that I think she was asking was why do I care about links?
The short and simple answer is: links from other sites to yours increases your sites value in the search engines.The more links to a page, the more value that the engine sees in your page.
Now I can give you a much more technical reason answer. But in the end the above is the reason links matter. The more links the more value. Better yet, the more valuable links to your page, the more value that your page will have.
You can build the most gorgeous, SEO'd website. Without links to the pages of that website, the site in affect will not exist in the engines eyes.