Psalm 51:16-17 meaning

Ritual worship, as we see it in the Old Testament among God's people, typically involved sacrifice. The people would bring something of value (such as an animal or produce) to the place of worship as a sacrifice to God. Incense would sometimes be used with a burnt offering to signify a pleasing aroma to God, in hope that He would receive the sacrifice and forgive the sins of the people (See Leviticus 1, 4, 5). 

David would have taken part in worship like this throughout his lifetime, so a Hebrew might be startled when they read David's words in verse 16, 

For you do not delight in sacrifice, otherwise I would give it;
You are not pleased with burnt offering (v 16). 

David also includes this idea in Psalm 40:

"Sacrifice and meal offering You have not desired;
My ears You have opened;
Burnt offering and sin offering You have not required."
(Psalm 40:6)

This Psalm is quoted in Hebrews 10 as being part of a statement Jesus made when He entered the world as a human (Hebrews 10:5-7). 

Samuel spoke this same idea to Saul when he disobeyed God's command, giving the excuse that he had saved back some of what was commanded to be destroyed in order to sacrifice to God:

"Has the LORD as much delight in burnt offerings and sacrifices
As in obeying the voice of the LORD?
Behold, to obey is better than sacrifice,
And to heed than the fat of rams."
(1 Samuel 15:22)

It appears that the Spirit used this lesson through David to prophesy the very words of Jesus when entering His fallen creation, in order to redeem it. 

New Testament believers can apply the words of David as seeing through the eyes of the newly created heart. We can then see what Jesus tells us in Matthew 5:8, "Blessed are the pure in heart, for they shall see God." 

If God does not want sacrifice and He is not pleased with burnt offering, what does He want from His worshippers? 

The next verse answers this question,

The sacrifices of God are a broken spirit;
A broken and contrite heart, O God, You will not despise (v 17).

God desires that His worshippers have a heart that follows Him—that seeks to follow His ways. Because our hearts have sinned, the only way we can have a pure heart is for it to be humbled and broken to the point that it contritely confesses its sin. 

If we do this, God is faithful and just to forgive us our sins (1 John 1:9). Here, David expresses this same sentiment when he writes: A broken and contrite heart, O God, You will not despise. As previously stated, God desires that we follow His ways; the ways that lead to life. But in order to accomplish this, we must first break the habit of following our own ways. We must first have a broken heart over our sin. 

This mirrors a theme that runs throughout scripture, that faith contrasts with pride (Habakkuk 2:4). A broken spirit is a spirit that gives up on the idea that "I know what is best for me, I don't need to listen to you." This is a spirit we are all born with; it is a spirit manifested by any three-year-old. The path to faith (that God knows what is better for us than we do for ourselves) is through a broken and contrite heart. This is a heart that is willing to see reality as it is. It is a heart that will listen to God (who knows all) rather than demanding to be God (which is an illusion). 

David is keenly aware that he is more than just a body existing autonomously in time. His heart and spirit—all that is most important about David, his relational commitments and spiritual orientation, his soul—best define who he is. The psalmist is a member of the human race, created in God's image with a soul purposed for eternal and loving companionship with his Maker (Ecclesiastes 3:11). That men and women have the capacity to act as free agents within the created order has often resulted in the false impression that individuals are accountable only to themselves. This is a grave mistake (Galatians 6:3). This was David's great self-deception, an error leading to an obstinate spirit of stiff-necked arrogance that offends God (Exodus 33:3; Jeremiah 17:23).

Jesus says in Matthew 5:3, "Blessed are the poor in spirit, for theirs is the kingdom of God." A broken spirit, contrite heart, and "poor in spirit" reflect the same idea. It is someone who recognizes the reality of the finite limitations of human reason and frailty, and who turns to God for direction and strength. 

To be "poor in spirit" means to be humble. To seek to serve God (and others) rather than yourself. It is the opposite of being proud or self-serving. The Greek word translated "Blessed" is "Makarios" (G3107). "Makarios" describes a complete and total fulfillment in life. It does not refer to a passing happiness or good fortune. It is an enduring state or condition that is unassailable.

The reason the poor in spirit are "Makarios" (happy) is because they embrace the reality of who they are and who God is. And they seek to live in harmony with God, which leads to our true fulfillment. The poor in spirit recognize their own brokenness (from sin) as David does within this psalm. But the poor in spirit also recognize our own human frailty and need to depend upon God for all things (John 15:4-5) as David's descendant, Jesus, did throughout His entire earthly life—but perhaps most clearly in the Garden of Gethsemane (Luke 22:40-42). As the poor in spirit do this, they actually become all they were intended to be, which leads to joy. 

Being "poor in spirit" is the absence of any pride in ourselves before God when confronted with our sin. For David, the cry for a "steadfast" and "willing" spirit is a recognition to continue with a broken spirit and a contrite heart. Worship will help him to keep the right perspective, God's perspective, in all that he does.

When we think of worship, our thoughts might go to singing songs in Sunday morning church services. For David, for Paul, and for followers of Christ, it is much more than that. Worship is living a life that acknowledges reality—most particularly God's sovereignty over all that is. 

Jesus points to worship as being this comprehensive concept in his conversation with the woman of Samaria: 

"But an hour is coming, and now is, when the true worshipers will worship the Father in spirit and truth; for such people the Father seeks to be His worshipers. God is spirit, and those who worship Him must worship in spirit and truth."
(John 4:23)

We also see scripture speak of worship in this manner in Matthew:

"And a leper came to Him and bowed down before [worshipped] Him, and said, "Lord, if You are willing, You can make me clean."
(Matthew 8:2)

Here the leper is recognizing Jesus as God, acknowledging His power (truth), and submitting himself to it—all an acknowledgment of reality as it is. 

The Apostle Paul, in his letter to the Romans, is helpful to point us in a direction of a lifestyle of worship. He says in Chapter 12:

"Therefore I urge you, brethren, by the mercies of God, to present your bodies a living and holy sacrifice, acceptable to God, which is your spiritual service and worship."
(Romans 12:1) 

Here Paul asserts it is reasonable to sacrifice ourselves to God, to live our lives fully to please Him. Paul's "Therefore" refers to the arguments of the previous chapters in Romans, that assert that to live fully to please God leads to life, fulfillment, and immense reward. To live for self, the flesh, and the rewards of the world is to squander life; it leads to death, loss, and slavery/addiction. When faced with the question whether we prefer life or death, the reasonable answer is "life." Paul argues that to gain life requires living as a sacrifice—one whose actions are pleasing to God. 

We are to take our everyday life—rising up, going to work, going to school, going to the gym, going to the store, interacting with our family, whatever else we do—and present it to God as a living sacrifice of worship. This is the path of our greatest benefit in life. When we look at our life from this perspective, we can see the coming together of the new, clean heart (Psalm 51:10) that David speaks of; a redemptive and worshipful heart.