They don’t call him Crash for nothing. In every level you must destroy boxes, and many enemies need to be defeated. The most basic attack is the spin attack. Press the Attack button, and Crash spins around with his arms out. This smashes most boxes and defeats most enemies. Combine it with a jump to hit boxes up high, or for a flying spin attack.
Jumping is another way to attack. Several enemies have no protection on top, so jump onto them to defeat them. Press the Action button while jumping to have Crash perform a Belly Flop (also called a Body Slam). This attack inflicts more damage and allows Crash to destroy stronger reinforced boxes.
The last attack uses the slide. Run toward an enemy and slide into them. This works as long as they don’t have spikes or other pointy bits sticking out to the sides that can cause damage.