What did the apostle Paul mean by the greeting, “Grace to You” in many of his letters?

"Grace" is an attribute of God that is manifested by kindness and favor shown to all people in some measure, but especially toward those that receive salvation. Grace is similar to forgiveness in that a debt is released (that debt being sin), and similar to mercy in that God has shown restraint in punishing sin. (Rom. 4:8) However, Grace goes far beyond these acts. In showing grace, God not only withholds judgment, but showers the objects of His grace with abundant favor and blessing. By showing grace, however, sin goes unpunished for a time. Because of God's perfect righteousness, justice must be served, and all sin punished, or it could be said of God that He was not good. (Prov. 17:15) God's solution was to allow His son, Jesus, to take the wrath that was rightfully due to the objects of God's grace. This was accomplished when Jesus submitted himself to a sacrificial death by crucifixion. Because of the incalculable worth and sinless perfection of Jesus, this was an acceptable sacrifice for the sins of the whole world all throughout time. Jesus was raised from the dead, proving he was truly God, revealing the Father's acceptance of His sacrifice, and defeating the curse of death.

The "grace" that Paul speaks of, is a general petition (prayer) that God would continue to show this grace upon those that Paul is writing (almost always being the true believers in Jesus Christ.) Even though a Christian is forgiven of his or her sins (justification), there is a life-long battle with the desires of the flesh, which often causes a believer to fall into temporary sin. So there is a sense in which a Christian is being shown continual mercy and grace (withholding of wrath for sin.) Paul often encourages believers to be ever-mindful of the grace to which they were shown and to behave in such a way that shows a humble appreciation of the magnitude of the mercy and kindness that God granted to them.