Thank you for the question on the Psalms, “Who was the Book of Psalms written for?”
The Book of Psalm was written for the priesthood of Israel to lead the people in worship and or repentance before God.
As one reads the “introduction” of each psalm like Psalm 30, one finds an introduction to the Palm: “A psalm. A Song for the Dedication of the Temple of David” (NIV). David wrote many of the psalms. When the Levitical priests before the Ark and later before the Temple, they would sing praises to God or lead the people in procession to worship God. The Psalms had music to them, but now the melody has been lost whereby the Psalms record only the words.
I Chronicles 16 says this, “He appointed some of the Levites to minister before the ark of the LORD, to extol, thank, and praise the LORD, the God of Israel: 5 Asaph was the chief, and next to him in rank were Zechariah, then Jaaziel, Shemiramoth, Jehiel, Mattithiah, Eliab, Benaiah, Obed-Edom and Jeiel. They were to play the lyres and harps, Asaph was to sound the cymbals, 6 and Benaiah and Jahaziel the priests were to blow the trumpets regularly before the ark of the covenant of God. 7 That day David first appointed Asaph and his associates to give praise to the LORD in this manner:
“8 Give praise to the LORD, proclaim his name; make known among the nations what he has done. 9 Sing to him, sing praise to him; tell of all his wonderful acts.
10 Glory in his holy name; let the hearts of those who seek the LORD rejoice.
11 Look to the LORD and his strength; seek his face always. 12 Remember the wonders he has done, his miracles, and the judgments he pronounced, 13 you his servants, the descendants of Israel, his chosen ones, the children of Jacob.
14 He is the LORD our God; his judgments are in all the earth. 15 He remembers his covenant forever, the promise he made, for a thousand generations, 16 the covenant he made with Abraham, the oath he swore to Isaac. 17 He confirmed it to Jacob as a decree, to Israel as an everlasting covenant: 18 “To you I will give the land of Canaan as the portion you will inherit.” 19 When they were but few in number, few indeed, and strangers in it, 20 they wandered from nation to nation, from one kingdom to another. 21 He allowed no one to oppress them; for their sake he rebuked kings: 22 “Do not touch my anointed ones; do my prophets no harm.”
23 Sing to the LORD, all the earth; proclaim his salvation day after day. 24 Declare his glory among the nations, his marvelous deeds among all peoples. 25 For great is the LORD and most worthy of praise; he is to be feared above all gods. 26 For all the gods of the nations are idols, but the LORD made the heavens. 27 Splendor and majesty are before him; strength and joy are in his dwelling place. 28 Ascribe to the LORD, all you families of nations, ascribe to the LORD glory and strength. 29 Ascribe to the LORD the glory due his name; bring an offering and come before him. Worship the Lord in the splendor of his holiness. 30 Tremble before him, all the earth! The world is firmly established; it cannot be moved. 31 Let the heavens rejoice, let the earth be glad; let them say among the nations, “The LORD reigns!” 32 Let the sea resound, and all that is in it; let the fields be jubilant, and everything in them! 33 Let the trees of the forest sing, let them sing for joy before the LORD, for he comes to judge the earth. 34 Give thanks to the LORD, for he is good; his love endures forever. 35 Cry out, “Save us, God our Savior; gather us and deliver us from the nations, that we may give thanks to your holy name, and glory in your praise.” 36 Praise be to the LORD, the God of Israel, from everlasting to everlasting.
Then all the people said “Amen” and “Praise the LORD.” 37 David left Asaph and his associates before the ark of the covenant of the LORD to minister there regularly, according to each day’s requirements. 38 He also left Obed-Edom and his sixty-eight associates to minister with them. Obed-Edom son of Jeduthun, and also Hosah, were gatekeepers” (A searchable online Bible in over 150 versions and 50 languages)
It is likely that the people began to memorize the words of the psalms as in many Christian churches, worshipers knows the lyrics by heart. They can sing the songs acapella or with music. The difference is that Psalms were used for the worship while in our modern contemporary music there is the usage of hymns and praise songs that may last a few years and are replaced by others songs.
SUMMARY: The Psalms were written for the worship team of the Levitical Priests who would put the words into music. The Israelites would learn the words of the psalms and the melody from the Levitical priests.
For more perspectives:
https://www.quora.com/Who-was-the-Book-of-Psalms-written-for