There are a number of reasons for the US population to increase to the point eclipsing many European Countires. Immigration is one of these. As the US expanded westerward and found itself possesing lands far in excess of the population needs it was able to accept immigration from primarily European areas. Europe throughout the 19th century continued to be a class dominated system, and the ability for land ownership as well as upper mobility was difficult for most. The opportunities for land ownership and a perception of equality of all men was attractive for those willing to make the transatlantic journey. Events such as the Irish potatoe famine, continued lack of opportunities for those of dissident religious or social groups also helped to encourage people to seek immigration. While there were period where immigrants were looked down on, through the 19th century the US continued to accept immigrants with limited screening. Indeed during the Civil War the need for soldier grew so severe that the Union advertised for soldiers in places like Ireland, where citizenship and opportunity were seen as an acceptable risk to service in the army.
Secondly, in comparison to European nations the US had a better quality of life for most people, especially as related to diet, exercise and cleanliness. The better diet allowed for increased sexual desire, increased fertility and increased the chances that children would survive to maturity, and would usually be healthier and larger than their European equivalents. An example of the summation of this idea can be seen in World War 2 when American GI's began arriving in Britain in preparation for European theatre opperations. A common refrain from the time, by the Brits was that the American GI's were "over paid, over sexed and over here."
Finally, the US became seen by many as a land of opportunity with various gold rushes, stories of rags to riches, unlimited opportunities fueling the mythology of the American Story, and persuading many people who would have otherwise stayed put making the move over to be part of the excitement then evident in the nation.