Increase in dogs and cats perhaps seen climbing (6)
I believe the answer is:
stepup
'increase' is the definition.
(I've seen this before)
'dogs and cats perhaps seen climbing' is the wordplay.
'dogs' becomes 'up' (I can't explain this - if you can you should believe this answer much more).
'and' says to put letters next to each other.
'cats perhaps' becomes 'pets' (cats are often kept as pets).
'seen climbing' is a reversal indicator (in a down clue, letters climb up).
'pets' back-to-front is 'step'.
'up' put after 'step' is 'STEPUP'.
'in' is the link.
Can you help me to learn more?
(Another definition for stepup that I've seen is " Increase".)