The CEFF web site will give you all the info you need. One of the CEFF's main purposes in life is to produce the national teams for the various disciplines eg rivers, lough style etc at home international, european and world levels.
For the rivers there's about 5 or 6 eliminators fished around the country each year- details on the web site. The top 32 rods nationally from these eliminators go on to fish the National Rivers Final and the top six rods from there will then constitute the following years Home International English Rivers Team. Continued good performances at a national level then make eligibility for the European or World Team a possibility.
The basis for each eliminator is usually (but not always) along the following lines:-
i) a draw to divide the entries into two halves and more specifically pairs of anglers
ii) one member of each pair will fish in the morning session (the draw dictates which one) and the other in the afternoon. Anglers are usually free to fish anywhere within the specified beat subject to certain rules about proximity to other anglers and time spent at any one point- again all on the website in detail
iii) the non fishing partner will control (umpire for want of a better word) the angler who is fishing i.e measure and record the number of fish caught and keep an eye out that the rules are being adhered to.
iv) points are awarded for each qualifying fish caught (only trout and/or grayling over a certain length depending upon the venue and season) Typically 20 points for each fish caught above the minimum length plus one point for each cm the fish measures.
v) the most points win
vi) swap roles for the afternoon session, which may or may not be on a different beat
vii) results from the two sessions are usually kept separate and the top rods from each session go through to the national final.
You've missed all the action for this year as only the final remains to be fished on the Tees in September but i can't recommend giving it a go next year highly enough. Win or lose it'll do amazing things for your own fishing as you'll pick up plenty by controlling for the top guys for three hours- you get a birds eye view of everything they do

Good luck for next year hopefully
CJ