Since 1776 deals mainly with military events, the only way to answer this question the way it was asked is to include McCullough's earlier, companion work, John Adams, which deals with the floor debate over independence. I will, thus, list the most important non-military events in 1775 and 1776 as covered by both books.
1) The Proclamation of Rebellion. The American Revolution broke out spontaneously on April 19th, 1775, at Lexington and Concorde when the British attempted to seize a store of arms, but were resisted by colonial militia. The British were forced back to Boston after intense colonial attacks as they retreated from Concorde. The Massachusetts militia, despite the pleading of representatives like John and Samuel Adams, lacked support from the Second Continental Congress as the moderate delegates hoped to effect a reconciliation. On June 17th, the Battle of Bunker Hill marked a significant escalation, producing 1500 casualties on both sides. Following the battle, Congress addressed the Olive Branch Petition to King George III, reaffirming loyalty to Great Britain and pleading with the King to intercede. Up to this point, many in the colonies blamed Parliament alone for the Crisis and felt that the King would protect them if he only knew the American side of the conflict. George III forever dispelled this hope when he issued the Proclamation of Rebellion on August 23rd, 1775. The King declared the colonies to be in open rebellion, labeled the Congress a "conspiracy of ill-designing men," and authorized the use of military force to end the rebellion. For many delegates, independence became inevitable at this point, as did a protracted war to achieve it.
2. Common Sense: What the Proclamation of Rebellion did for the delegates in the Second Continental Congress, Common Sense did for the common people. Published by Thomas Paine on January 10th, 1776, Common Sense was a treatise arguing for American independence from Great Britain and the establishment of a single American nation with an egalitarian government. Clear, concise, and powerfully written, Paine's message spread like wildfire throughout the colonies, encouraging those outside of Boston to make common cause with the Massachusetts militia.
3. The Evacuation of Boston: After Bunker Hill, the British stayed within Boston while George Washington, now in command, struggled to bring order and discipline to the volunteer force. Thus, neither side actively sought battle. This stalemate was broken when Colonel Henry Knox pulled off a logistical miracle, dragging the heavy cannons from Fort Ticonderoga in the wilderness of upstate New York over mountains, lakes, and rivers to Boston. Washington then, quickly and in great secrecy, had the cannons placed at Dorchester Heights. When British General William Howe saw that his fleet had moved within range of heavy guns overnight, he knew that his strategic position had become untenable. British barrages were unable to reach the Heights. Washington received word that the British would spare Boston if they could depart unmolested. On March 17th, Howe and his forces departed for Britain. The Evacuation of Boston was a crucial intersection between the military and political situations, similar to battles like Saratoga and Trenton. The British retreat was stunning, and it showed the colonies that military victory was possible.
4. The Lee Resolution: Richard Henry Lee of Virginia arrived at the Continental Congress on June 7th, 1776, with new instructions from his home-state of Virginia. Convinced by the victory at Boston that the Revolution could be won, leaders in Virginia authorized Lee to propose the following resolution:
Resolved, That these United Colonies are, and of right ought to be, free and independent States, that they are absolved from all allegiance to the British Crown, and that all political connection between them and the State of Great Britain is, and ought to be, totally dissolved.
This was a crucial moment in the drive for independence within Congress. Up until this point, Massachusetts had been the driver of events, and it was crucial that the other colonies also take a stand. Virginia, as the most populous colony, was the natural leader of a nation-wide independence movement. In fact, it was, in part, because of Virginia's prominence that George Washington had been named the leader of the Continental Army.
5. The Declaration: The Lee Resolution produced a furious floor debate between moderates and revolutionaries. Despite the Proclamation of Rebellion, many in the colonies were not yet resolved to breaking with Great Britain. The wealthiest colonies in particular—New York, South Carolina and Pennsylvania—were unwilling to risk the damage to trade and property that a war of independence threatened. Making matters harder, John Adams and the other leaders of the faction for independence recognized that the Lee Resolution must be passed without opposition, or not at all. On June 11th, the debate ended, and a final vote was scheduled for July 1st. A committee was also formed to draft the Declaration of Independence, should the motion carry. Slowly, after seeking support from home, the delegates privately assured Adams of their colony's support until only New York, Pennsylvania, Delaware, and South Carolina remained in opposition. At the same time, Jefferson worked on the declaration, drawing upon rhetoric he had used many times before in other documents and pamphlets. South Carolina was finally persuaded. Caesar Rodney rejoined Congress to push Delaware into the independence column. The most recalcitrant delegates from Pennsylvania, recognizing they were outnumbered, agreed to absent themselves from the final vote. New York, meanwhile, facing a massing invasion force on Staten Island, promised to abstain from the vote provided there was no opposition. On July 2nd, the Lee Resolution was carried 12-0. On July 4th, independence was formally declared and the final version of the Declaration of Independence was published and read aloud throughout the colonies. The United States of America had its independence—if it could win it.
Aftermath: In military terms, the remainder of 1776 was a disaster. Washington was unable to prevent the British landings at Brooklyn and Manhattan. His forces were then driven from New York City, defeated at Fort Washington and White Plains, and chased all the way to Pennsylvania. The Declaration of Independence seemed like a hollow gesture in the face of such failure. As December 1776 neared its close, Washington faced the disintegration of his army as enlistments were nearly up for many soldiers.
In the face of this nadir, Thomas Paine again took up his pen and published The American Crisis, which successfully rallied public opinion and the attitudes of many soldiers to continue the fight. In order for this new resolve to be sustained, however, Washington needed a victory. Thus, on Christmas night in 1776, the American commander famously led his troops across the icy Delaware River and launched a successful dawn attack on the Hessian troops at Trenton. This victory was followed up by another surprise attack at Princeton. Washington's dauntlessness convinced the army, the colonies, and the world at large that the cause was still alive.