1 In reference to the new lands discovered
When the land was newly discovered, the name was initially given as 'The Americans'. Sooner or later, the United States utilized the name of the continent to refer to its new country.
2 South America kept this name
Various South American 'Spaniards' gave this country's name as America, that represented the country after Columbus discovered the entire land.
3 People were influenced to call themselves Americans
When people started landing on this newly discovered land, they were influenced by the government of the United States, that they should establish their identity as 'Americans' and start calling them by this name. This way, they could achieve a unique identity as that from Britishers.
4 Britishers gave this name
The name ‘America’ emerged when British people were offered to live on the new continent ‘The Americans’. They started calling the continent as ‘America.’
5 The US independence made the name official
The US declared independence in 1776. Consequently, the official name was introduced as the USA ' United States of America,'. (See Why is the American Flag Red, White, and Blue?)
6 The US was the first independent country in America
Just because the US was the first country in the new continent 'America' became independent, they were exclusively allowed to add 'America' to their formal name 'US.'
7 The word colonists need to be changed
The US citizens needed to come upon another name to determine their nationality. Since the US's independence, the citizens were stopped being called as colonists, and recognized as 'Americans.' (See Why the United States is called a nation of immigrants?)