This term is a bit more complicated than merely "smoking opium". It starts when you have your first high, the world is peaceful, everything is perfect, you're numb, but in the best way possible. But, soon, it starts wearing off. Fast. Your mind races, you're pulled out of your dream world. You crave the drug more and more, wanting to feel the same way as you did on your first high. You go to the dealer and buy the same amount you had the first time, and smoke. Still feels good, but not as good as first time. You go and buy more. Closer, but not quite there. You're stuck, you don't know what to do. You want to go back to that little dream world and stay forever, but your body is already developing a tolerance. You panic. You use all your money to buy more and more and more, but still, not the same as that first time. You realize that you have no more money, so you start selling your things, pawning whatever could get you that next bag. Still, nothing compared to what you had on that first, magical time. So, you're broke and own nothing. But you don't care, all you care about is getting back to the first high. You start stealing, doing "favors", whatever gets you the money for the attempt. Your life becomes a living hell, all in search of a repeat of the first high. That's chasing the dragon.
I doubt an example is necessary here.