I'm playing this game with the speedrun in mind and trying to go fast but I can't seem to figure out the tilt mechanic.

Mechanic is pretty simple, you press B to tilt the right side of the table and Left on the DPad to tilt the left side of the table.

The tilt is a very effective tool to get the ball into the correct positions, e.g. for skipping from the middle to the top in the Poppy stage or for the "tilt bash" quick kill on Wispy.