I Should Tell You...

I think it's about second chances

"Don't let her down" He's saying that to himself...

I think there's a possibility that he's relieved and overjoyed that he has received a second chance with this person.

"I watched the falling rain...splash upon your face...I know those aren't your tears..."

I think that means he's thinking "Hey buddy it;s cool. She;s not pissed or upset...you STILL got a chance..."