This is as simple as the description says. Just do 5 Quick Turns (click both analogue sticks and you'll see your character jump up and land on the opposite side of the Katamari while making a whooshing sound) before you finish the level. I did it exactly 5 times and got the trophy to unlock. Make sure to be extra careful that you don't do it a sixth time or the trophy might not unlock (not 100% sure on this yet). I suggest waiting until time is almost up to do the 5 Quick Turns and then waiting until the level is over.
The trophy will unlock once you Trophy Update is finished and you're returned back to the Village Square.