The enamel on the inside of the skillets is satin smooth, so that food does not stick to it, so the first thing to do is make sure you do not mar the smoothness of the porcelain. I recommend cooking with a wooden or nylon spoon, do not use metal as it has a tendency to scratch.
When washing do not use steel-wool or anything abrasive.
If you let food scorch and burn on the bottom, add some baking soda to water, put it in the skillet, and then put it on the stove on a low heat. All the burnt food should come off. Then clean as normal with washing up liquid and a soft sponge. Then dry the skillet and put some white vinegar on a kitchen towel and wipe it over the enamelled surface. It should come up satiny. Wash and dry again, and your skillet is good for another cooking session.
Of course the easiest thing is not to let your food burn in the first place! For normal cleaning, just soak in warm soapy water and then hand wash with a sponge. The same cleaning technique applies to Le Creuset Dutch Ovens.