The most likely is that when you play in the company of other top players, you are forced to cover more distance, sprint more, run faster. The overall level of the match demands that you perform at a higher physiological level, and drags you up to that level. There are other studies, for example, that confirm this, by showing that when players from these lesser leagues play against players from top leagues, they must run more and faster than they are accustomed to.
So now, the implication should be clear if you are playing in the World Cup, against some of the greatest players in the world, at the highest level of competition, the physiological demand is maximal (as it would be for Champions League, Id imagine). Under these circumstances, the risk of fatigue is greater than ever you take a player who is accustomed to running 2km fast, with 400m sprinting in 75 sprints, and you force them to run 2.5km fast, and sprint 100 times to cover 650m, and that player would struggle over 90 minutes. The fatigue effect, the drop off in sprint performance is thus likely to be even greater. It is the same as saying to a 10km runner who is accustomed to running 3:00/km that they have to start at 2:50/km. By 7km, the effects will be clear!
And this is why physical conditioning is so vital to elite teams. Ultimately, I would be overstating the value of sports science (I am biased, after all) if I said this was decisive to the outcome of matches. Its not, and there are so many other factors that determine the result. Physical conditioning is but one of them. But what I can say is that if players are NOT conditioned for the demands of the match, then their decline in performance may cost them.
And finally, remember that it doesnt take much to be shown up by an elite player if you concede even 1m over a 20 m sprint (5%), then you look like a carthorse alongside a thoroughbred! And fatigue will cost you that 5%! So next time you are watching a match, and you suddenly start seeing players leaving others behind (whereas at the start, it was always an equal contest), you may realise that this could be due to a shift of even 1m over 20m, 5%, and a goal that wins the game may be the result!