As long as I can remember, the biggest clubs typically field teams of reserves mixed with youth in the early rounds. They may take it a bit more seriously as they advance depending on how congested their fixture list is, and their league standing.
It's an entry into the UEFA Cup or Euroleague or whatever the heck it's called these days, so there is an incentive for clubs that are mid- to lower-table.
edit: jeez, I basically repeated the first reply
Otherwise, it's just an excuse to field a rotating or younger squad. It isn't a fully 'open' competition like the FA Cup. The FA Cup is about 90-100 years older and it comes with Europa qualification.
Supporters of the big six could not care less about it.