Music in general has the ability to calm an individual; it has a therapeutic power.1 If the evil spirit caused Saul to feel depressed,2 then it makes sense that a servant would propose some music. Additionally, it was a common view amongst the cultures around Israel that music warded off evil spirits.
The real reason that David could bring relief to Saul, was because the relief was accompanied by Bible truth. When Saul’s servants are singing David’s praise, they mention his ability to speak well. In the context of playing the harp, that implies that he sings while he plays. He sings songs of encouragement, songs that are full of truth about God. Songs that would remind Saul of God’s promises. Even the songs we find in the Book of Psalms. David thus brings Saul into contact with God’s Word. He comes to do good for a man under the curse. The promises of the gospel continue to be heard by and held out to the rejected king.
16 Let our lord now command your servants who are before you to seek out a man who is skillful in playing the lyre, and when the harmful spirit from God is upon you, he will play it, and you will be well.”