Psalm 51:13-15 meaning

After pouring out his heart to God, confessing his sin, and expressing confidence in God's mercy to forgive his sin in verses 1-12, David asked to have the joy of his salvation restored, and to be given a "willing spirit" that followed God's ways (Psalm 51:12). He now makes a practical statement as to the benefit that will flow from God giving David a "willing spirit" that will follow in His ways: Then I will teach transgressors Your ways (v 13). 

This would indicate that David's teaching to transgressors would come largely through example. As David's willing spirit follows God's ways, observers can see the benefit of walking and living in God's (good) design. It is in this manner humans can gain their greatest fulfillment in life. 

David has recognized that he is a sinner himself (Psalm 51:4). But he also advocates to God that by restoring him, and giving him the wisdom of a "willing spirit," David will become an example. Not only will transgressors be taught, but also sinners will be converted to You (v 13). 

This "willing spirit" that David prayed for is an inner wisdom, a renewed mind that recognizes that our true self-interest lies in walking in God's ways. It is through such a renewed mind that we can be transformed, and walk in the newness of life (Romans 12:1-2). 

New Testament believers have the indwelling Holy Spirit of God that is always leading us. But we also still have our old nature, what the Apostle Paul calls "the Flesh." These two forces contend within us, each seeking to persuade us to follow in its path (Galatians 5:17). God leaves it to us to choose which to follow. We can, like David, pray for a "willing spirit" that will choose to follow the Holy Spirit (Psalm 51:12). Then, also like David, we can be an excellent example to others. 

David had many responsibilities in life, including his family, his calling and his vocation as King of Israel. As king, David was the leader of the government as well as the spiritual leader for his people. While all of these roles were important, they were to be a part of his ultimate purpose which was to be redemptive in each of these "jobs." 

The steadfast and "willing" spirit to follow God's ways for which David asked is the inner wisdom to make good choices—choices to follow in God's (good) design for our lives. When we follow in God's ways, we bring redemption to a fallen creation by restoring small pieces of it to God's original intent. We bring harmony in place of dissension. 

Just as our physical heart needs to beat in rhythm for our body to function properly, our new divinely created heart needs to beat in the proper rhythm, which David now realizes is a redemptive beat as he says, 

Then I will teach transgressors Your ways,
And sinners will be converted to You. 

By living an example of walking in harmony with God's ways, others are brought into those ways as well. In this manner, we can bring redemption to a fallen world. 

The example of confession and repentance David explains and demonstrates for us in this Psalm is a necessary discipline for us to exercise in order to consistently live in harmony with God. Peter describes this as "sanctifying Christ as Lord in your hearts" (1 Peter 3:15a) and keeping "a good conscience" (1 Peter 3:16). 

When we live in harmony with God, by confessing our sins like David did, it also provides an occasion for transgressors who are hurting or jaded to pause in wonder about our hope. It may even prompt them to ask us "to give an account for the hope that is in" us (1 Peter 3:15b). As we do this, we too can say with David: Then I will teach transgressors Your ways, And sinners will be converted to You. This is, therefore, an important equipping in order to carry out what is often called the "Great Commission" of making disciples as we go about our lives (Matthew 28:19-20).

In the prior section, David had asked to be given a "clean heart" and a "steadfast spirit" (Psalm 51:10). Paul asserts that for New Testament believers, each person who believes in Jesus is given both: 

"Therefore if anyone is in Christ, he is a new creature; the old things passed away; behold, new things have come."
(2 Corinthians 5:17) 

As a result of being a new creation, each believer has the great opportunity to live a redemptive life, and bring a bit of restoration to a fallen world by walking in God's ways. We can do this by choosing to walk in the Spirit, following God's ways (Romans 8:4). 

The thread of a new creation being the very heart of living a redeemed life runs through Psalm 51 as well as 2 Corinthians 5. In 2 Corinthians 5:15-16, Paul asserts: 

"and He [Jesus] died for all, so that they who live might no longer live for themselves, but for Him who died and rose again on their behalf."
(2 Corinthians 5:15) 

This is consistent with Jesus's teaching that the way to gain the greatest fulfillment in life is to lay aside our lives (Matthew 16:25). The broad road of following our own desires and appetites leads to destruction. But the narrow road of following God's design leads to life (Matthew 7:13-14). 

David says that a by-product of living in fellowship with God is to teach transgressors Your ways, And sinners will be converted to You. Paul says the same thing a different way in 2 Corinthians, calling this ministry of conversion "reconciliation"—speaking of reconciling a fallen humanity back to God's (good) design for creation:

 "Now all these things are from God, who reconciled us to Himself through Christ and gave us the ministry of reconciliation, namely, that God was in Christ reconciling the world to Himself, not counting their trespasses against them, and He has committed to us the word of reconciliation."
(2 Corinthians 5:18-19)

Paul goes on to classify the calling of New Testament believers as ambassadors to bring reconciliation to the world. But this only happens if we live in a manner as being reconciled to God ourselves:

"Therefore, we are ambassadors for Christ, as though God were making an appeal through us; we beg you on behalf of Christ, be reconciled to God."
(2 Corinthians 5:20)

For Paul, as for David, this purified heart would reconcile him to God, then create reconciliation to those around him, through his testimony: Then I will teach transgressors Your ways, And sinners will be converted to You.

David realized that there was another part to the new beat of the "clean heart" that God had created in him through forgiving his sin (Psalm 51:10). In addition to bringing reconciliation to those around him, David would exercise vocal worship of the Lord God: 

Deliver me from bloodguiltiness, O God, the God of my salvation;
Then my tongue will joyfully sing of Your righteousness (v 14).

The use of the word bloodguiltiness by David likely refers to a confession to his shedding of innocent blood, the blood of Uriah, the husband of Bathsheba. David ordered that Uriah be put in front of the battle, where he would be killed. To add insult to injury, he knew of Uriah's loyalty, so he sent the message by Uriah's own hand (2 Samuel 11:14-15). 

Continuing his prayer from verse 12, Restore to me the joy of Your salvation, David declares in verse 14 that my tongue will joyfully sing your righteousness. But then David adds: O Lord, open my lips, That my mouth may declare Your praise (v 15). David apparently had been unable to praise God while he was steeped in sin and guilt—understandably so. It seems here that David is asking God to put praise back in his mouth, to open my lips. God had already spoken forgiveness to David through Nathan the prophet, telling David that he would not die, but also had told David of consequences that would ensue from his transgression (2 Samuel 12:13-14). Here David is taking his forgiveness to a deeply personal level.

This might be a picture of what it looks like for a New Testament believer to approach God's throne, seeking a cleansed conscience, as spoken of in Hebrews:

"Therefore, brethren, since we have confidence to enter the holy place by the blood of Jesus, by a new and living way which He inaugurated for us through the veil, that is, His flesh, and since we have a great priest over the house of God, let us draw near with a sincere heart in full assurance of faith, having our hearts sprinkled clean from an evil conscience and our bodies washed with pure water."
(Hebrews 10:19-22)

Since David had plumbed the depths of his sin and appealed to God, the God of his salvation, he prays for God's provision to put into his mouth words that will match that depth with the heights of his public declaration of the greatness of God.