Along with bringing the French into the war as full allies, up until that point they had been supplying the Colonial forces with weapons and ammunition, the battle of Saratoga also marked a huge upswing in popular support for the war. From the very dark days of the encampment at Valley Forge, Colonial fortunes had been steadily improving and the battles of Freeman's Farm and then Bemis Heights set the stage for General Burgoyne's surrender. This surrender and the subsequent expansion of the fight against the British by the French in the West Indies and elsewhere began to lead the British to feel that their effort to maintain their control of the colonies simply wasn't worth the cost in men and equipment and money.
The change in the momentum of the war was arguably the most important result of the battle. The massive influx of militia from the request by colonial governors saw for the first time the British being both outfought and out-led by their colonial counterparts. This excitement and positive momentum combined to create the circumstances that would build the confidence and the strength of the colonial army to a point where they could defeat the British strength for strength.
The most significant thing about the Battles of Saratoga is that the American victory in those battles led the French to fully enter the war on the American side. For this reason, the battles are seen as a turning point in the war.
Before Saratoga, the French did not want to commit to the American cause completely. They did not want to be completely committed to the Americans unless they had a good chance of winning the war. This victory convinced the French that it would be possible for the Americans to win. Therefore, they entered the war, helping the Americans immeasurably.