Could you explain to me the meaning and usage of the phrase
"to be cool with something"?
Where and when would it be appropriate to use it?
The idiom to be cool with a thing means to not find the thing objectionable.
"Would you be cool with dropping by the post office on the way there?"
"Sure, no problem."
"I thought you would be cool with inviting her. I didn't realize you had some feud going on."
Please be warned, that American slang usage of "cool", "hot", and "lukewarm" to indicate enthusiasm or agreeableness can be complicated and at times contradictory. To say that someone is "cool towards" someone else means they are expressing disdain; to say someone is "cool with" someone else means they don't find that person objectionable, and will get along collegially with them.