Turns the page = Moves on, takes us to the next part of our lives.
"Like the shadow" is a simile but the line after that gives us the key to its meaning: "I swear like the shadow that's by your side I'll be there." He will be there like a shadow, (perhaps only on sunny days or in brightly lit rooms).
Sometimes people cry when they are very happy, especially women.
Student or Learner