I am curious as well, so I did a (very) little bit of prodding and poking about on the internet. Best I can figure is you have to be on a team . . . for me it would be team USA. I imagine you need to participate in some qualifying events, starting with your local FITA shoots, and moving up to Regional, State and National competitions. Once you are at the state and national level, I am supposing you can either request an invitation to join the team, or request a tryout, or perhaps it works more like basketball . . . they scout you and pick you if you are the best. I figure the very best way to get there is to shoot everything, post AWESOME scores, and get noticed. I really think the State and National FITA shoots are where you want to be . . .
Here in Utah, it seems that all things FITA go through the Pro Shop Salt Lake Archery. They have a (slightly) outdated website, but per chance could give better info over the phone. Out of state, I would try contacting the larger pro shops and see if any of them know what's going on

The Utah shop is
www.saltlakearchery.com, hope that helps!