Before David's encounter with Goliath, he was a musician. In fact, he first came to the royal court to soothe King Saul's mental distress with the beautiful melodies of his harp. Bible depictions of him as a musician help to support the traditional view that David was also a prolific poet.
Nearly half the poems in the book of Psalms carry an attribution to David. And a prayer of David that appears in Second Samuel 22 also shows up in Psalm 18, indicating that he could have authored this and other psalms. Texts outside of the Biblical canon also claim that the total number of psalms David wrote tallies in the thousands.
Some modern scholars, however, see it differently. They say the attribution—which can mean "to/for David"—is really a dedication to David, celebrating his life as a great king. At any rate, these attributions, which appear as superscriptions above those psalms, were probably later additions.
But it may not be much of a stretch for Bible readers to believe that David was responsible for the poetry of those Psalms once they feel the soulful experience of his life through the Biblical prose.
