Here are the USEF selection procedures for 2012. Note on page 6 that "The number of invitations for United States representatives to the 2013 FEI World Breeding Dressage Championships is determined by the FEI and WBFSH." I don't remember how many invitations were granted to USA in 2012, and don't know if that will change for 2013, but Hallye Griffin at USEF should be able to fill you in.
Bah, I'm not from the US, so this doesn't apply. I was hoping someone from the UK or something would chime in.
Since competitors are selected by their national federations, you will want to talk to your own NF about its procedures. The 2013 dates for the championships are Aug 7-11, so qualifying has to be done early in the year. For instance, the USEF selection trials are on May 23-26 (Western Selection Trials), and June 6-9 (Eastern Selection Trials).
I did find this, regarding the 2013 championships:
"Following lots of considerations and discussions, the F.E.I. determined the selection criteria so that there will not be more than 50 competitors in each age-group (five- and six-year-old horses)."
Each National Federation gets a designated amount of slots. It is up to the NF to determine their selection process. So you will need to check with your NF. A quick search of your NF or email to them is probably much quicker and more accurate than asking here since the majority of us posting are USA folks.
Golden State - 2012 Bundeschampion & 2014 USDF Horse of the Year
Golden West - 2014 & 2015 Bundeschampion Pony Stallion
Petit Marc Aurel- FEI Dressage Pony Stallion