Something that's furnished is supplied with whatever it needs. A fully furnished kitchen has every appliance and ingredient you need to cook meals for yourself.
When you rent a furnished apartment, it comes will all the furniture you need, including a bed, chairs, table, dresser, and sofa. If your sewing kit is furnished, it's got all the needles, thread, pins, and scissors you require. When you're not describing a home, furnished means "equipped," from furnish, from the 15th century meaning "equip or provision," or "provide to soldiers."