1. Backgammon is a great way to teach math to your children. And the amount and complexity of the math will increase as their age and skills increase. When they are very young, they will simply learn to add the numbers on the two dice and count the pips going around the board. Eventually, they will get into far more complex math including odds and statistics, first by understanding something as simple as there are more ways to get hit when you are 6-away than when you are 1-away, and then, eventually, learning much more complex concepts such as winning odds and take points.
2. Your children will learn about luck. About how to handle good and bad luck with grace. Learning to cope with dancing on a one-point board, or losing a tough game will help them learn to cope with the bad luck and misfortunes that life throws at all of us. And learning to win with grace and good sportsmanship, if you are careful to teach that to them (primarily through example) is equally important in life.
3. Learning to play a complicated, adult game like backgammon will help build a child's confidence and self image. It will give them something to be proud of, to feel special about.
4. The backgammon community provides a social outlet for children. There are many other opportunities for children to meet and interact with other children, but backgammon offers a unique opportunity for children to meet and interact with adults on an equal basis. This is particularly true if you involve the child in live tournament play.
5. Backgammon gives the child a hobby that he can enjoy virtually any time on the computer. And it is a far better hobby and diversion than mindless video games, watching cartoons, and many other diversions kids use to waste time or relax.
6. Backgammon gives the child a pastime that he can share with the entire family. And it is possible for reasonably young children to be very competitive with adults, so it is not always about the child simply learning from the adult. And if you enjoy going to tournaments and spending time playing backgammon, wouldn't it be much more enjoyable if you could bring your children with you instead of having to spend your leisure time apart? And even now that my sons are grown, I have the pleasure of playing doubles with my sons and we've even traveled together to tournaments around the country.
7. And lastly, backgammon is a game that the child can enjoy the rest of his life. How many kids spend hours learning cheerleading, playing hockey, and doing other sports and games that are worthwhile and fun while they're young, but once they become an adult, it is not likely that all those hours of practice and honing their skills will be useful. And the younger they start and the more they learn when they are young, the better head start they will have at being better players when they are an adult.

I have to add a word here about the negatives. What is wrong with children learning backgammon? Why don't all parents encourage and teach their children to play? About the only major negative I can think of is that many parents associate backgammon with gambling, and they don't want to expose their children to tournaments and web sites that focus on the money and gambling aspects of the game.

My response to that is in modern society children are constantly exposed to the fact that there is a lot of wagering and gambling going on. What child doesn't see all kinds of ads for the lottery; for casinos and Las Vegas? Kids are interested in the Super Bowl, but there's hardly a report without giving the point spread.

The media is full of poker games, horse racing, off-track betting, and there are dozens of TV shows where contestants try to win big money and prizes every day. So your kids are going to be exposed to gambling anyway. Isn't it better for them to see that it is possible to compete for the joy of winning and for the trophy, and for the pleasure of doing something well? These are things you can teach your children by focusing on these aspects of the game when you are with them.

And let's never forget what Fran Liebwitz said about playing games with children: "I love to play with kids—they're easy to cheat and fun to beat!"