(American slang:)
To retrieve, usually in an unhurried or lackadaisical manner.

For example, the golf ball shagger mentioned above is a device to aid in the retrieval of golf balls. Similar devices exist for tennis balls. I recently overheard a grocery store manager telling an employee to "go out into the parking lot and shag some carts." Not knowing (or recalling) the British usage (see iain's writeup above if you're unfamiliar with it), neither supervisor nor underling found anything unusual in their dialogue, but I had to chuckle.

The best known instance of this American usage of shag comes from baseball. During batting practice, many balls are hit into the outfield and must be retrieved. Since the balls are usually hit on the fly, they're called "fly balls" or simply "flies." The resulting expression "shagging flies" seems perfectly innocuous to the average baseball player assigned to the task, but quite hilarious to someone with a different mental picture of the activity.