As Anthony said, the payment is for the subscription of The Daily Prophet. We see this happen a couple times in the books.
‘Hagrid!’ said Harry loudly. ‘There’s an owl –’
‘Pay him,’ Hagrid grunted into the sofa.
‘He wants payin’ fer deliverin’ the paper. Look in the pockets.’
Hagrid’s coat seemed to be made of nothing but pockets – bunches of
keys, slug pellets, balls of string, mint humbugs, tea-bags … finally,
Harry pulled out a handful of strange-looking coins.
‘Give him five Knuts,’ said Hagrid sleepily.
‘The little bronze ones.’
Harry counted out five little bronze coins and the owl held out its
leg so he could put the money into a small leather pouch tied to it.
The Philosopher's Stone - Chapter 5: Diagon Alley
Hermione, however, had to move her orange juice aside quickly to make
way for a large damp barn owl bearing a sodden Daily Prophet in its
‘What are you still getting that for?’ said Harry irritably, thinking
of Seamus as Hermione placed a Knut in the leather pouch on the owl’s
leg and it took off again.
The Order of the Phoenix - Chapter 12: Professor Umbridge
It is not mentioned every time a subscription of The Daily Prophet shows up because it doesn't need to be. Rowling established that you pay a few (between 1 and 5 apparently) Knuts for the subscription, and trusted that the reader would remember that. This information is worldbuilding, but not really integral to the plot, so it's not mentioned every time.
The rest of the time we see owls being used as messengers, it's either Hedwig/Pig whose job is to do whatever Harry/Ron wants. Or it's a school owl that also has to do what they want. The owls themselves could not care less about the money. They just want food and care. So long as they get that, they seem to be willing to ferry mail around.