Forgiveness of sins, freedom from guilt
The meaning behind the power of the gospel is that it alone can bring forgiveness of sins.
Acts 13:38 Be it known unto you therefore, men and brethren, that through this man is preached unto you the forgiveness of sins:
The power of the gospel offers freedom from sin, not a license to sin. Along with the freedom from sin, comes the freedom to serve God.
Romans 6:22 But now being made free from sin, and become servants to God, ye have your fruit unto holiness, and the end everlasting life.
The power of the gospel also offers freedom from guilt. It means you don't carry a debt load of sin anymore.
Colossians 2:13-14 And you, being dead in your sins and the uncircumcision of your flesh, hath he quickened together with him, having forgiven you all trespasses; Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to his cross;
Praise the Lord, that he nailed every sin that was against us to HIS CROSS!
There is therefore now no condemnation to them which are in Christ Jesus!