It's the saying of the Golden Snitch. It simply means that at the "close", the "ending", it would open. When Harry would truly need it, to decide the war, it would open, containing one of the deathly hallows, which is the Resurrection Stone.
(To that ^^^ person, I believe something else.)
I believe that 'open at the close' means he's ready to die. The close of his life, and by walking towards Voldemort, he's opening (starting) it.