I raise my own rabbits, and most feeds found at farming/feed stores are primarily hay (and some grains/veggies). Rabbits do have fat, but it's on the skin. You'd have to request that and render it for food usage.
My rabbits are on an organic diet of orchard grass hay, and an oat hay/timothy hay pellet mixture. They also get cuttings of non-sugary vegetables such as kale, spinach, celery, lettuce, various grasses, oats, parsley, mint, rosemary, thyme, etc. For treats or when the does give birth, I give them carrots, berries, apple cuttings, and beets.
The meat is very high in protein, much higher than any other animal of which I am aware. They are low in fat, and are very low in saturated fats, unfortunately, but there are benefits of rabbit meat. They have 100% of the daily recommended dosage for vitamin B12 in about 3.5 oz. of meat and they have a good deal of digestible iron. They're low in both cholesterol and calories, and are by far one of the cleanest meats to consume. Very few diseases and bacteria in rabbit are transferable to humans. They don't receive vaccinations or antibiotics in most rabbitries. Their livers are fabulous for pate! [http://www.threelittleladiesrabbitry.com/blog2/2011/01/20/nutritional-value-of-rabbit-meat/]
My favourite rabbit dish is French dish called "Lapin a la Coccotte": http://www.food.com/recipe/lapin-a-la-cocotte-french-rabbit-stew-114799, which is very similar to Coq au Vin, another family favourite.
If you worry about the low fat content, it's encouraged to have desert afterwards. I like pairing my rabbit with an avocado "pudding".