Contrary to what many believe, it's not Independence Day of Mexico, but a day to celebrate Mexico's victory over France at the Battle of Puebla
nteresting is that it gets celebrated more grandly in the United States
You are correct, it was not Mexico's independence day, which was 50 years earlier.
Cinco De Mayo gained notoriety in the US primarily because teachers started putting up posters and had kids do artwork, in elementary schools to celebrate Cinco De Mayo. It quickly caught on and became a big day to party on college and university campuses throughout the US.
Notwithstanding, the date does have real significance in the US.
Mexico did not just defeat the French but caused the French to leave North America. Had they not, they most likely would have supported the confederate states during (and after) the American Civil War and could have altered the outcome as we know it.