The simple explanation: You have fish and you have plant(s). The plant lives in some sort of container similar to what you might normally expect (except not using the soil you are used to). The fish live in a different container. The plant get's it's water from the fish container. Plants need food to live, which they also get from the fish water. The fish need clean water. This process of the plant eating and drinking from the fish water cleans the water too! It's clean, organic, natural and fun. The fish do need their own food, but this is usually as simple as giving them appropriate fish food.
For a more technical description, aquaponics is the blending of aquaculture (fish-keeping) and hydroponics (growing plants without soil). In an aquaponic system the fish, plants, and various bacteria form a symbiotic group: the fish eat food and excrete waste. The fish waste is consumed by the bacteria which break it down into nitrates, which are absorbed by the plants. This cycle helps clean the water of waste products, which keeps the over-all system habitable.
Why should you choose aquaponics as a method of gardening?
The short answer is that you can have great results with little resources and effort. Aquaponics uses less space and much less time than traditional gardening. It can use ten times less space than normal gardening, which means you can have ten times more (great) plants in your small garden than would be possible in a normal garden. Weeding, tilling, digging and all the other headaches that go along with soil maintenance go away. Try something new, you will most certainly be happy with the results.