The Declaration of Independence helped the American colonies break away from Great Britain in at least two ways.
First, the Declaration helped to rally support among the colonists. We do not always realize this, but not all of the colonists were interested in rebelling against Britain. Scholars estimate that even during the Revolutionary War, only about 1/3 of colonists really favored independence. They believe that 1/3 opposed it and the remaining third did not really care one way or the other. By laying out the case for independence, the Declaration swayed some people to the side of independence, helping the rebellion to succeed.
Second, the Declaration helped to rally support among foreign countries. The colonies were likely to have a very hard time fighting Britain on their own. They needed financial, military, and diplomatic support. The Declaration helped them to get this because it helped convince other countries that the colonists’ cause was just. This was a factor in getting France to provide vital assistance to the colonies.
In these ways, the Declaration of Independence helped the rebelling colonists get support both from their fellow colonists and from foreign countries. This support helped the colonies successfully break away from Great Britain.