The short version:
Points are awarded for placing 1st through 5th. Five points for 1st, four points for 2nd, and so on. Additionally, twenty five or more teams doubles the points awarded and fifty or over triples the points. For example, a third place finish in at a contest with 63 teams is worth 9 points.
Your best five top five finishes are added together to equal your total points in a category OR overall. However, any of these five categories ( chicken, ribs, pork, brisket, overall) have no correlation to each other in any way. You could be top five in each of the four categories and not even be in the top five for team of the year (just an example). One note, if you have less than five scores to add, obviously they only use the scores you have.
I hope I've explained this in a way that is easy to understand. I reread it and I just don't know how to put it better, even though it could easily be misunderstood.