You celebrate the day that the U.S. got it's freedom and was no longer under control of the British. Also, I think its when the U.S. became not a part of the 13 colonies anymore.
Every country has their own independence day and they celebrate it from different countries. Ex: for the US it is July 4th, independence from the United Kingdom
July 4, because the Declaration of Independence from Great Britain was signed on that day in 1776.
In American date it would be, 07/04 as they swap the month and date around, compared to the British...