Yes, that is how it happens. When you are eligible (3 years GC holder) you simply apply for American citizenship. You do not have to do anything else if/when you become an American citizen- you are AUTOMATICALLY a dual. You do not have to inform Canada or anything, your Canadian citizenship is not in danger. Many, many people are dual Canadians/Americans (such as Alanis Morissette and Celine Dione. )
Where people get confused is in the oath ceremoney you swear that your allegiance is to the US and and only the US or something like that but they CANNOT force you to renounce your citizenship, are you are free to become an American citizen so long as you meet the 3 year GC requirements.
There are a few countries that don't allow dual citizenship, like Japan, but the US is not one of them.
