Items can only be used in Campaign Mode.
Before each campaign battle, you will see a screen where you can choose which items you want to use.
You can only use one specific item of each type, but you can use multiple types of items at the same time.
Also, items are only consumed if you win a battle, so do not worry about wasting items if you lose a stage.
Be sure to select "Start Battle" to activate the items after selecting them.