If we are regenerated children of God and led by His Spirit, we will exhibit His character and spiritual image. Before God summoned us, began to reveal His truth to us, forgave us, justified us, and imparted His Spirit to us, our spiritual father was Satan! We were no better than the Pharisees, whom Christ told that they were of their father, the Devil, because they were doing Satan's works (John 8:39-47). Children display the characteristics of their parents, so Christ judged the Pharisees to be the children of Satan because they were exhibiting the Devil's characteristics.
Before God intervened in our lives, we, too, were the children of Satan (Ephesians 2:1-3) because we were exhibiting his spiritual characteristics. However, God began to redeem us and called us into a relationship with Him, which, as Romans 8:15 says, was symbolically an adoption. God was not our original father, but He took on that role after He extracted us from the grasp of Satan, sin, and this world.
Verse 16 reiterates that the Holy Spirit is intended to provide a witness of who we are and who God is. If we allow the Spirit to lead us, we are sons of God. It follows that, if we are sons of God, then we will be exhibiting the same characteristics as our Father! When we exhibit God's characteristics, we are a witness to the world of His character and the way He lives.
Under the New Covenant, with access to the Holy Spirit, the quality of our witness must be much higher than what God expected of physical Israel. To whom much is given, much also is required (Luke 12:48)! If our neighbors, co-workers, or family members look at us, and all they see are people who go to church on different days, do not eat certain foods, give multiple tithes on their income, and do not believe in the Trinity, are they seeing anything different than Old Covenant Israel, who did not have the Holy Spirit? Certainly, God's law will set us apart from the world because the world is against God, but merely keeping the letter of the law will not provide the complete witness that God is looking for.
This is not to denigrate the royal law of liberty to any degree. Acts 5:32 says God gives His Spirit only to those who obey Him. However, one can be nominally obedient, keeping God's law in the letter, without making a truly effective witness for God.