Something that's full-fledged is completely developed, or fully qualified. A full-fledged meal, as far as your mom's concerned, might include meat, potatoes, and a vegetable.
Things that are whole or complete are full-fledged — a full-fledged case of the flu isn't just a few sniffles and coughs, but requires you to lie in bed with a fever for several days. A full-fledged war involves two countries with a conflict deep enough to inspire violent clashes on both sides. The fledge part of full-fledged comes from the Old English -flycge, "having the feathers" or "fit to fly."