Hair definitely grows from the roots. Once it is out of your head, a hair is basically dead, and the ends or any part of it do not grow. The only growing part is at the roots. Each hair root, or follicle, contains cells that spin out long microscopic fibres at a very rapid rate (at least at the microscopic scale). These fibres then twine together into larger fibres which twine together again to end up creating a growing hair. The person who told you about cutting off split ends just gave you the wrong reason. The reason for cutting off split ends is to help your hair become more manageable and incidentally to look nicer. Split ends do not have anything to do with hair growth.

More information on hair can be found at L'Oreal. In particular the parts "Portrait of an Unknown Element" and "Superb Chemistry" within that.