You qualify for the Champions League (or Europa League, the one just below it) that takes place during the next season. So because of Arsenal's finish in the BPL last season, they are playing in the Champion's League this season (specifically the playoff to get into the Champion's League proper, which starts with the Group Stage later on, since they finished 4th in the BPL). It's a strange mechanic for those used to other sports, since this year's teams will be playing in the Champions and Europa leagues even though it was last years teams that qualified (though admittedly very often not that much changes).
It's one thing a lot of people don't take into consideration with soccer, especially the good teams that make deep runs into these non-league commitments. Real Madrid last season, for instance, played their league games, all of the Champions League games (since they won it), and their own internal league cup. Good teams from other leagues go through the same thing - it's way, way more games than just the league games, at least for the top teams in the world.
So a lot of strategy comes into play regarding when you can rest players on the easier games so they can be fresh for the big ones, etc. The best managers walk a very thin line with which players to play and when. It's also why the really, really good teams require so much money - their rosters are essentially packed with top players, a lot of them often sitting the bench as "super subs." So when a team like Real Madrid subs in players for a game to rest them, their enormous financial resources allow those subs to actually be good enough to start on most other club teams. That allows the top, top teams to be competitive across multple commitments outside their leagues, which lesser teams simply can't do.
And of course, you also have players with international team commitments, too, which only adds to their workload on occasion.