This is an important question because if you live in town and don't already have a lot of tools and accessible land, you may discover that it's cheaper to buy your food in supermarkets than to grow it. You need answers to this question other than "To save money..." Consider these answers:
There's no better way to learn about nature. The garden is a kind of ecosystem in itself. The garden has its own system of energy flow and it recycles its nutrients such as nitrogen. When you see wasps collecting spiders from your squash plants you'll understand more about predator/prey relationships than if you read a whole chapter on it. Soil has its own ecology and if you don't understand it your plants may die. If you don't understand that those pretty cabbage moths leave behind eggs from which hatch caterpillars that eat your cabbage before they metamorphose into new cabbage moths, you may lose your cabbage crop. Every garden is a zoo and a botanical garden where you can see thousands of different kinds of organisms -- different families of them, different genera, species and subspecies. All the basic elements of nature exist in any average garden.
Fresh food is healthier to eat and it tastes better than store-bought, processed food. When vegetables are harvested many nutrients in them begin breaking down. The longer the period between harvest and being eaten, the more those nutrients are lost. Moreover, just look at all the chemical additives added to your foods to preserve them, color them and add taste. Fresh food grown by you and not contaminated by chemicals is much better for you and tastes better than average store-bought food.
Working in the garden is healthy for you. With all its stooping, digging, hoeing, carrying, etc. it provides exercise you may not get in your everyday life. However, maybe the most healthy thing it does for you is to give your mind and spirit a moment to commune with Nature! I can't explain how it happens but I can promise you that just working a while in the garden each day raises your spirits as it tones up your body.
Gardening helps the Earth heal. Sometimes it seems that everything we do ends up killing something nice or destroying a part of Nature we don't mean to hurt. Gardening gives us a chance to help Nature restore herself. The plants we grow convert industry's carbon dioxide to oxygen, which we animals need. As we add organic material to soil as we garden, the soil's structure improves and its microflora and -fauna becomes more healthy. Any garden increases species diversity on the ground it occupies. Knowing that you're doing at least something to help the Earth heal can make you feel a lot better. Finally, when we eat something from our own backyards we're not paying people to transport produce from distant parts of the world, keeping it cool all the time, with all the carbon dioxide and other pollutants that entails.
Gardening is fun and even therapeutic. Each morning you walk out and see how things have grown, what's been nibbled on during the night, what might be picked that day... You just walk around and it's like visiting an old friend who has changed overnight! Moreover, when you fill your head with thoughts about flowers and fruits, soil, bugs, weeds, tools, fertilizer, etc., those thoughts replace gloomier, negative thoughts. The garden heals the sad, unhappy spirit!
And it CAN reduce your food bill. Most gardens in towns and suburbs are more hobbies than serious efforts to produce food for the daily table. However, if you really grow what you want to eat, preserve those things with canning and freezing, and truly eat these things instead of going out or eating processed food, the savings can be substantial.
classic summer squash