Possibly the hardest part of this level is the Tap-Tap egg shot. You want to jump towards the left side of the second platform (the one just below the red star flower), and start aiming an egg during that jump. You want to aim the egg just long enough for the aiming cursor to travel a few blips up, and then release as you're landing on the platform. Hold right the entire time, including to fall onto the stone area where the tap-tap was. Jump on this stone area to travel over the tap-tap. A faster way to do this is to shoot a horizontal egg much earlier (see IL record), which allows you to avoid the flower and save 0.5s on the score screen.
The movement and camera scrolling starting at 0:18 is important to make sure the moving platforms load in optimally and the stair cloud is on screen so you can hit it as early as possible. When you hit the stair cloud, the camera will lock in place, and if Yoshi touches the edge of the screen after this you will immediately lose control. Because of this, try to keep the screen from going too high before hitting the cloud or you will lose control when you land on the ground where the spear guys are. A short jump off of the last moving platform (when you shoot the stair cloud) should keep the camera from going too high.