In December of 1776, George Washington was facing defeat in the Revolutionary War. He had been retreating through New Jersey since losing Long Island and New York City earlier in the year. By December, his troops in the Continental Army were cold, hungry, tired of the long retreat, and ready to go home when their enlistments were up at the end of December. At this point, Washington took a huge gamble. On Christmas night he and his troops crossed the Delaware River. On the morning of December 26 he routed Hessian troops fighting for the British at Trenton. Washington was victorious at this battle. On January 3, 1777, he met British troops at Princeton where he again was victorious. He then wintered in Morristown, New Jersey. These two small victories raised the spirits of the Americans and kept them fighting. All the British had to show after nearly a year of fighting was control of New York City.
Also in December of 1776, the Continental Congress fled to Baltimore, fearing an attack on Philadelphia. This allowed the new American government to continue functioning.
This was a particularly low point in the American effort for independence, in that we had faced a long series of military defeats, and Washington's army was a shadow of its original size and quality. Most of America's major cities had been recaptured by the British by that time and it looked like the Revolution might indeed fail.
The less than 2000 soldiers that remained, however, were veterans, and determined, committed troops. Through sheer force of will, they beat the Hessians at Trenton, and another British force at Princeton shortly after. This raised morale, gained badly needed supplies and helped to recruit fresh troops.
From the civilian side, Americans both willingly and unwillingly contributed food, money and supplies to the American revolutionary effort, provided intelligence information of British troop movements and strengths, and often cared for the wounded soldiers. What's not as well known is that a large number of Americans also remained loyalist to the King and provided the same assistance to the British side.
Since your question refers specifically to The Crisis by Thomas Paine, published in 1776, his efforts as a writer who could distill the principles of the cause down into simple and understandable phrases for the average American and soldier helped to unify the army around the cause, and to understand what they were fighting for.
Among his more famous passages:
These are times that try men’s souls. The summer soldier and the sunshine patriot will in this crisis, shrink from the service of his country; but he that stands it NOW, deserves the love and thanks of men and women.