Ok, first off, this isn't ELI5, this is AskHistorians, and there is a assumption (hope?) that you've done at least a basic bit of googling/wikipedia reading first. However...

Nobody "owned" West Germany, West Germany was the part of Germany occupied by the US, France and England after WWII. East Germany was the Soviet occupied half. Berlin was split between the two occupying forces. Eventually both occupied sides of Germany formed their own governments. Needless to say, East Germany formed a communist government, while West Germany became a western style democracy.

East Germany, like pretty much every communist totalitarian state ever, did not want people leaving the country, because in short order, the best, the brightest, and the common working class, and pretty much anyone else who could, would leave for the higher quality of life in one of the western democracies, thus leaving the communist state lacking in both an educated class, and a working class. This is widely considered a Bad Thing(TM) among nation states.

This page should give you some more in depth details. Also, Wikipedia has some basic information that should help you as well.