Affiliating has two important goals, one short and one long and I will explain each below:
Long Term Goal: Pagerank, pagerank, pagerank.
Our almighty and all knowing lord, google uses over 200 variables in their formula to determine how popular your site is on the search engine and one of the few variables that we mere mortals are allowed to know about is called pagerank. It goes from 0-10 and twitter is a 9 last time I checked.
To get a higher pagerank you need good quality links pointing to your site(this is why spam bots go and spam links on sites so Google indexes them and then their pagerank goes up).
When an affiliate has their link area on every page, google considers that a brand new different link pointing to you each and every page. (This is why by default links in signatures in most forums have nofollow)
The higher the pagerank is, the more your site will popup in the engine and your traffic will eventually fuel it self.
Short Term Goal:
Obviously indexing links and the such takes google a while but there are immediate rewards to affiliating and that is local traffic from the other site instead of Google.
Though it can backfire quite often if you affiliate with a site that is in the same niche as you.
Lets say I am a member of Forum A which deals in games and that board affiliates one day with Forum B which also deals with games. As a member I will not go to Forum B because I already have an awesome gaming forum.
Example you want:
As a member of gaming Forum A which deals with games, I notice that one day they affiliated with Forum B which deals with RPing/Writing. I decide to check it out and register and end up liking it. Now I have a forum dedicated to gaming and one to RPing/Writing.
This is also why a bunch of times out there you will be told to focus on one niche because chances are someone will always be doing something better then you.(Can't be a super mega board for all types of things) Those have the hardest time getting traffic.
It also helps to affiliate with sites bigger or similar in size to you.
As an example my RPing/Writing site affiliated with a tech site and a warcraft3 gaming site and increased my site's pagerank to 3 within the first 3 weeks of purchasing the new domain.(It takes sometimes 2+ for sometites with 100+ affiliates to even get that far because they typically do it within the same niche and google will more often then not just mark it as similar content and not really promote yours.
I hope I was able to educate some users and if you want to affiliate here just post about it, maybe we can get a ring going of good quality affiliations.
My site is up for affiliating if you are of similar size: http://www.rpdom.com