how did bilbo find the way out of gollums land

Bilbo used his new ring to stay invisible and was able to follow Gollum to the exit. He had to leap over Gollum and charge through the door. His buttons got stuck and he had to leave them behind to get away. The orcs had seen his shadow and he had no time to extricate himself before they got him. So he just pushed through and lost the buttons.

Bilbo realises his life is in mortal danger and runs through the maze of pitch black tunnels as Gollum chases him. Bilbo trips, and suddenly finds the ring on his finger, but believing he has no chance to escape his pursuer, he stays where he is and prepares to meet his fate, but Gollum runs right over him. Bilbo realises the ring makes him invisible. He manages to escape past Gollum, who has gone to guard the only exit, and finds his way to the surface where he rejoins the Dwarves.