"America" is short for "The United States of America". It's also the only shortened name which doesn't need to be preceded by 'the'. And it's the only single word which can be used to describe people of the US so then it's also been used to refer to the country. Originally 'American' was used to refer to people of British America, an area comprised of the eastern half of present-day US and Canada. Later the term came to refer to just the US.