Green manure is not the same as raw manure. Green manure, sometimes alternatively called cover crops, is when a crop or plant is grown and then intentionally plowed under in order to improve the soil. Green manure may also be obtained by collecting green leaf and twigs from plants grown in wastelands, forests and other areas, although in the case of organic farming, growing green manure crops is more common.
Cover crops used for green manure include soybeans, clover, rye and others. The main benefit of using green manure for organic farmers (and other farmers) is that adding organic matter to the soil releases important plant nutrients, including nitrogen and helps to increase crop yields. Other benefits include improved soil condition, decreased weeds, increased water holding capacity and decreased soil loss via erosion.
Bonus points go to green manure because it's an inexpensive way to improve land and it's an eco-friendly farming practice that helps decrease emissions that would add up if you had to transport fertilizers to the farm.
If you're looking for ways to incorporate green manure into your organic farm see the following guides: