how did the goals of labor unions in the late 1800s reaffirm the ideals expressed in the Declaration of Independence?

The Declaration of Independence proclaimed fundamental values and principles of organizing society, in particular by stating that the Laws of Nature should govern mankind.

Fundamentally, providing for the creation of the social environment in which people can live with their unalienable rights guaranteed and protected. Meanwhile, in a democratic society it is the task of people themselves (via Labor Unions in this case) to self organize and make sure that these principles are followed through. In a broad understanding Labor Unions in the late 1800s were pursuing worker-specific rights for 'life, liberty and the pursuit of happiness' buy promoting minimum wages, fixed length of working hours per day, and improvement of working conditions.