I try this from my own word, haven’t checked into any commentary yet. Please correct me if I am wrong.
15 O Lord, open my lips,
and my mouth will declare your praise.
Self explained - the psalmist asks God to help him perhaps into an appropriate disposition so that he can praise God appropriately, that his praise may be pleasing to God.
16 For you have no delight in **sacrifice**;
if I were to give a burnt offering, you would not be pleased.
This verse is probably the contention: what is the sacrifice that the psalmist refers to here? I’d say it is the usual offering of doves, lambs, etc.; being offered to God as atonement for sin or to please Him. This verse is thus, seems to me, revolutionary especially at the time of the psalmist where such offering was rather the norm for worship. So what does he try to say?
17 The sacrifice acceptable to God is a broken spirit;
a broken and contrite heart, O God, you will not despise.
This is the answer for v.16. The sacrifice that is acceptable to God is a contrite heart, a broken spirit, rather than the conventional sacrifice they they used to offer Him.
The sacrifice of a contrite heart is a sure sacrifice that will be accepted by God. This is significant because many a times in the OT, God did not necessarily accept sacrifices offered to Him. Some He accepted, some He did not. Thus the often times pleas by worshippers or psalmists - ‘may my sacrifice be acceptable to You, that You will be delighted by it, and find favor with me’.
Later in the Gospel, Jesus mentioned about the poor in the spirit in the Beatitude, a person who hunger for God.
The psalmist’s prayer in Psalm 51 foreshadowed our prayer and practices today - to offer ourselves as a living sacrifice to Him by changing ourselves (repentance) which is made known by St Paul in Rom 12:1-2.