How did the european starling became the most common bird in america?

Like most introduced animals, they easily out-competed the other birds and were able to thrive in the new environment.

Are European starlings native to America? No, they are a European species, introduced here.  
How did they arrive at their present habitat? People brought them and released them where they are.  
How did they become one of the most common birds in America? They easily out-competed the others and were able to thrive in their new environment.  

Ballast Water- Organisms have been brought over by water, when waves pull them away from their original home. Two examples of this would be mussels, and maybe some fish.  
Ship Stowaways- Organisms and animals have been brought over by ships. They get on the ship and they are moved over here. Two examples of this would be a rat and maybe some spiders.  

I hope this helps.