Game associated with US includes quiet period of heightened activity (4,4)
I believe the answer is:
rush hour
'period of heightened activity' is the definition.
I can't tell whether this defines the answer.
'game associated with us includes quiet' is the wordplay.
'game' becomes 'RU' (rugby union).
'associated with us' becomes 'our' ('our' is the possessive pronoun for 'us').
'includes' indicates putting letters inside.
'quiet' becomes 'shh' (I can't explain this - if you can you should give a lot more credence to this answer).
'ru'+'our'='ruour'
'ruour' enclosing 'shh' is 'RUSH-HOUR'.
Can you help me to learn more?
(Other definitions for rush hour that I've seen before include "Traffic-snarl period" , "lot of coming and going" , "'Time of most road traffic (4,4)'" , "Time when traffic is heaviest" , "Busy period to travel" .)