Start a level and dash exactly 6 times before finishing the level. Dash by repeatedly wiggling the left and right sticks up and down (sticks should be wiggled alternately, so when the left stick is down, the right stick is up) After you wiggle them a few times, the katamari will spin and then go forward really fast and make a rushing sound. Trophy will not unlock until you finish the level and go back to the village square.
Dashing doesn't seem to work on Katamari Drive mode.