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.
We are challenged in the Word of God to put on a whole new garment or wardrobe. This wardrobe should match the new life that is in the Lord Jesus Christ.
Col. 3:12–17 Put on therefore, as the elect of God, holy and beloved, bowels of mercies, kindness, humbleness of mind, meekness, longsuffering; 13 Forbearing one another, and forgiving one another, if any man have a quarrel against any: even as Christ forgave you, so also do ye. 14 And above all these things put on charity, which is the bond of perfectness. 15 And let the peace of God rule in your hearts, to the which also ye are called in one body; and be ye thankful. 16 Let the word of Christ dwell in you richly in all wisdom; teaching and admonishing one another in psalms and hymns and spiritual songs, singing with grace in your hearts to the Lord. 17 And whatsoever ye do in word or deed, do all in the name of the Lord Jesus, giving thanks to God and the Father by him.