Example..."How you doing buddy? Great great, I asked my girlfriend to marry me and she said YES. That's awesome, you must feel fantastic! Oh man, I have the world on a string!"
Example..."I've finally started selling my paintings online and the sales are going GREAT. I sold two yesterday and four last week, and a guy called me and asked if I could do a portrait of his entire family and he offered an enormous sum of money. This is amazing, I feel like I have the world on a string."
Explanation - If you have a little puppy on a strong leash you will be in complete control of it. The puppy will walk where you want it to walk and stop when you want it to stop. The world on a string is an over exaggeration of that feeling. The world often feels big and dangerous and we don't usually feel in control of our lives, but sometimes everything just turns out great and we forget about our bad luck. In those moments we feel like the world is a little puppy we're taking for a walk on a string and it's doing exactly what we want. It's a feeling of "everything is just so great right now", like when you're falling in love. Like the song says...
It's a variant of the idiom "Has the world by the tail" and it means to be happy and often very confident, because everything is going the way you've planned it.
For example: He just landed his dream job, he has the world by the tail.