Newness of life is becoming a new creation or a new creature in Christ Jesus.
2 Corinthians 5:17 says that: 17 Therefore if any man be in Christ, he is a new creature: old things are passed away; behold, all things are become new.
This scripture is basically saying that a Christian should not behave themselves the same way that was done in their life before Christ. There should be a newness of spirit.
Romans 7:6: But now we are delivered from the law, that being dead wherein we were held; that we should serve in newness of spirit, and not in the oldness of the letter.