"The Battle of Bunker Hill" proved that the Americans could actually take on the strong, established British troops. The British did end up winning the battle, but ended up suffering many casualties. After such a hard hit, the British had to rebuild, giving the Americans a time to retreat and recuperate.
The importance of Bunker Hill was that the Patriots fought their hardest and they never gave up. This let the British know that the Americans wouldn't be easy to beat.
Furthermore it's worth stating that the British failed to exploit their costly victory and that led them to be forced of evacuating Boston on March 17,1776, which meant a decisive strategic victory for the American revolutionary forces.