Organic means the food was not grown with anything synthetic.
For meat it means the livestock was fed natural feeds only, and was not given any shots of antibiotics or growth hormones.
There is a serious health benefit to eating organic meat. If a cow was shot full of estrogen and then you eat meat from that cow, you're getting estrogen from that meat. Which is bad, very bad. This is also true for other animal products (milk/eggs/cheese).
Fruits and veggies though, organic means that they used natural fertilizers and natural (or no) pesticides. Fertilizer has no bearing on how healthy fruit is for you, and so long as you wash your fruit and veggies before eating, pesticides don't either. So in this regard there is no health reason for eating organic fruit and veggies, people who do so do it from a "save the planet" standpoint rather than a "save my waistline" standpoint.