I've tried to register several time over the last year and several of my competitions are listed with DMOZ.
It's likely that:
1. They are or know a Dmoz Editor; or
2. They paid a Dmoz Editor; or
3. They got lucky with their Dmoz submission!
1/2. Dmoz Editor:
Volunteer to become an editor for several Dmoz directories. You will be able to submit reviews of your own sites and have the opportunity to get to know your market better. That leads to more linking opportunities.
3. Dmoz submission:
a. Choose the correct category. Looking at potential categories before submitting your site. Find sites related to yours that are already listed in Dmoz, by using the search function.
b. Read Dmoz guidelines. Use a proper title and description. Use the name of your site; do not use anchor text. Do not use all capital case or promotional language (best, great, etc.).
c. Keep on topic. Think from the Dmoz Editor's perspective. If you're submitting your website as a golf magazine, but a big chunk of your articles are about weight loss, then the Editor is going to have a tough time categorizing it.
There's some great tips for Dmoz submission here:
Code:
http://www.kenkai.com/seo-blog-article-169.htm
Good luck.