Immigrant workers helped to achieve (and create) the American Dream in two major ways.
First, the immigrant workers helped to create and achieve the American Dream for themselves. These workers demonstrated the idea that people could come to America and pull themselves up by their own efforts. Their efforts helped to create the idea of the US as a land of opportunity and achieved the American Dream for themselves and their descendents.
Second, the workers helped to achieve the American Dream for the whole country. The work that they did helped to create a new, industrialized America where more and more people were becoming prosperous. This new prosperity (as seen in the 1920s) was, in a sense, evidence that the American Dream was being achieved.