This boss battle is against Whispy Woods. Stay on the far left side for the entirety of the battle. Whispy Woods begins the battle with a mask on, which must first take enough damage to be knocked off. Until this mask is off, he can't use his spit attack.
His three attacks are as follows. His first is dropping apples from his branches. These roll towards you and must either be jumped over or swallowed. The second attack is a spike attack. Three spikes appear, one after the other along the ground from right to left. If you're on the very left of the screen you are out of range of these spikes. If these spikes touch the apples, they will disappear.
Whispy Wood's third attack is his spit attack. He will spit out projectiles either at short range; at long range, curling upwards into the air; or at long range, curling downwards towards your position. These attacks are at random, so if the player jumps up into the air from the far left side of the screen each time a projectile is fired, regardless of its direction, Kirby will remain safe.
To defeat Whispy Woods, you must simply wait for apples to roll to you. When they're close enough, swallow them and spit them at the boss. Continue this, as well as dodging each attack, until he's defeated. The strategy is very simple for this battle so mess around if you wish to.
Once defeated, this level is done and you can move onto the next one! Jump onto the Warp Star and you'll appear on the level map. Move to the next level.