To add equipment and spells to your heroes you just need to tap the equipment on your rack and then tap the hero that you want to equip. It is important that you check the ingame help for details on the item you are equipping your hero as not all equipment is permanent, scrolls for example will only apply to the hero's next attack or heal. If your hero has spells you can cast them by tapping it and then tapping where you want to target it.