This is not a simple question to answer, not because it is difficult to come up with good reasons but mostly because it is difficult to express them all at the same time. Perhaps an explanation of what I believe the word organic signifies will be the best way to express its value.
Organic is not a process, it is not a particular way of doing things, it is not a set of standards to be followed, it is a holistic philosophy of food production and distribution.
The common way of doing things these days focuses on a limited and often backwards sense of value. Appearance, shelf life, global uniformity, monetary cost, and productivity factors are valued above nutritional value, health risks, and environmental and social impact. This backwards mentality produces farms with animals living lives any reasonable human would call inhumane in the name of cutting costs, plants that have been selectively bred or genetically engineered to produce more edible product per plant yet that are incapable of living or reproducing themselves without human intervention, plants that grow prettier fruit with fewer vitamins, chickens grown for meat that are physically incapable of supporting their own weight after a certain age, chemical fertilizers and pesticides that destroy the delicate balance of life in and above the soil, pollute the local water tables, runoff into nearby rivers and lakes, and make their way into the food we eat. This mentality treats animals, plants, and the soil itself as exploitable resources to be consumed without end. The results of this mentality are soil erosion, species extinction of both plant and animal life, the destruction of local economies and ways of life. These things cannot be replaced. Soil is built up inch by inch over thousands of years. Species and cultures once destroyed are gone forever.
Organic is trying to operate using true value, to work with nature rather than against her, to look at a bigger picture of sustainability rather than a narrow one of profit. Diversity is and has been the key to survival of life on this planet since the beginning of time. An intimate connection with the land has been the key to successful sustainable agriculture in the past.
Large global corporations are working against diversity and are attempting to sever the connection between consumption and production of food. They manipulate the media, government, and education in an attempt to control public policy and influence public opinion. They do not give this world what it needs but prefer to try to brainwash the world into believing that it needs what they have to give it.
The only way to fight against this is to become informed. Question everything you read, get both sides of a story, form your own conclusions and think for yourself. This all requires a different mentality and can be quite hard. With organic becoming a marketing label rather than a philosophy it is even harder. You have to question everything, read labels, research companies, find local suppliers, and sometimes go without some of the things you are used to eating. It's not easy but it is certainly rewarding to exercise another form of democracy: choosing where to spend your money.