“Say something, I'm giving up on you
Say something”

A Great Big World – Say Something Lyrics on LyricsMode
EXPLANATION
The repeating of "Say something, I'm giving up on you" is trying to show that he's coming to the end and, well, giving up. Then the last line where he says "Say something" is him giving up on them after fighting for it to work for so long. The way he says it is also important, he says it as if he was exhausted or tired. Which implies that he gave up a good fight that wasn't very successful.
EXPLANATION
Telling someone else to say something and using the I word is all about their narcissism. That isn't about communicating. It's all about the one person. Era of narcissism! Actions count. Words mean nothing.
EXPLANATION
All the guy wanted was for the girl to react to his declaration that he is giving up. In a way, the guy want the girl to ask him to hold on. Yes, he surely love the girl but there are many factors why he feels that what they have is hopeless. That's why he is giving up.
EXPLANATION
This relates to a person relationship with God. The person is desperate for God's answer and he is giving up and losing faith in God.