What is the meaning of the phrase "with all our spry young fellows painting the Old World red, and carrying off your best actors and prima-donnas"?...

There is a certain timelessness to Oscar Wilde's 1887 story The Canterville Ghost. During both World War II and the so-called "British invasion," there was a sense among many in England that its culture was being spirited away. During the former, American soldiers stationed in England in preparation for the Normandy landings married British women and returned after the war to the United States with their foreign brides. During the "British invasion," American culture was inundated with British influences in the form of The Beatles, Herman's Hermits and many other British bands. Even today, the British often complain when American filmmakers appropriate British history with fictionalized, Americanized, stories of heroism, as was the case with the films The Great Escape and U-571. In this sense, Wilde's novella was more than a little prescient. 

As The Canterville Ghost begins, Lord Canterville is showing his aristocratic family's estate to an American diplomat and the latter's family. During the course of their tour, Lord Canterville, displaying his sense of propriety, warns the American, Hiram B. Otis about the house's macabre history, explaining that the structure has long been haunted by a ghost. Otis's response to this information is revealing:

"My Lord," answered the Minister, "I will take the furniture and the ghost at a valuation. I have come from a modern country, where we have everything that money can buy; and with all our spry young fellows painting the Old World red, and carrying off your best actors and prima-donnas, I reckon that if there were such a thing as a ghost in Europe, we'd have it at home in a very short time in one of our public museums, or on the road as a show."

With his reference to "spry young fellows painting the Old World red," Otis is referencing the phenomenon discussed above: The "New World" Americans have persistently raided the "Old World" British for the latter's culture through the appropriation of many of Britain's more talented artists. It could be said that there was a tendency on the part of the British to 'look down their noses' at their American cousins, seeing the latter as a culturally inferior spin-off of the former. Hiram Otis is merely accentuating that perception with his comments. In other words, there is a very good chance that the American family moving into the Canterville estate will eventually return home to the United States and take the ghost with it.