The balloon went higher and higher until it finally disappeared from sight.
Literal
Balloon [topic-は] rapidly up went-and, finally invisible became.
どんどん上がって行って uses どんどん (steadily) + 上がっていく (rise and go away). とうとう means 'finally/at last.' 見えなくなった is 'became invisible.' Vivid description of a balloon floating away.