Why was the Battle of Bunker Hill important?

The Battle of Bunker Hill, though it did not accomplish anything strategically for either the Americans or the British, can be said to have had primarily a kind of psychological importance. Though the British had been defeated two months earlier at Concord, this had been one of those situations (which continued to occur throughout the war) in which they were unprepared for the unorthodox tactics used by the American rebels. The British were not exactly repulsed at Concord Bridge, but during the retreat to Boston, they were repeatedly ambushed by Americans firing at them from farmhouses and concealed positions all along the route. The British believed that if they could confront the rebels in open battle, in a situation more or less replicating European-style set piece battles, they would easily defeat them and quickly bring the rebellion to an end.
At Bunker Hill (actually Breed's Hill), this did not happen. The British had to make multiple charges up the hill, taking a shocking number of casualties, before finally making the American position collapse. This set a pattern, in a way, for much of the warfare over the next six and a half years until the final major confrontation at Yorktown. The British were able to defeat the Americans in open battle, but the cost of doing so was enormous. The British were not prepared to suffer such a high number of casualties or to make the enormous (and expensive, not simply in human terms) effort required to put down the rebellion. This was true especially in view of the fact that much of Parliament, and even many of the British military, were not enthusiastic about prosecuting the war in the first place. The Whig party was sympathetic to the Americans and believed that the rights of the colonists as fellow Englishmen had been violated. Eventually the war dragged on year after year with the rebellion no closer to being put down in 1781 than it had been in 1775.
Bunker Hill, though a British "victory," achieved nothing strategically. It merely inflamed the Americans even more and—given the bombardment that set the town of Charlestown into flames during the battle—demonstrated to them the ruthlessness with which the King and his administration intended to quash the rebellion. Within a year of Bunker Hill, the British were forced to evacuate Boston, with no end to the fighting in sight.