This verse is a pivotal moment in the story of David showing kindness to the descendants of his dear friend Jonathan. Here's a breakdown of its meaning:

"Mephibosheth, the son of Jonathan, the son of Saul, came to David..." This introduces Mephibosheth, establishing his lineage. He's the grandson of King Saul, who was David's predecessor and initially his enemy, and the son of Jonathan, who was David's loyal friend and Saul's son. This is significant because David had made a covenant with Jonathan to show kindness to his descendants.
"...and fell on his face, and showed respect." This describes Mephibosheth's posture and demeanor. Falling on one's face was a customary sign of profound respect and submission to a king. Mephibosheth is presenting himself humbly before David. He may also be showing deference and fear, as he is the descendant of a king David defeated.
"David said, 'Mephibosheth.'" David simply calls Mephibosheth by name. This is a personal touch, acknowledging Mephibosheth.
"He answered, 'Behold, your servant!'" Mephibosheth responds with a formal term of submission. By calling himself David's servant, he acknowledges David's authority and places himself at David's disposal.

In summary, this verse illustrates:

The fulfillment of a promise: David is beginning to fulfill the oath he made to Jonathan to show kindness to his descendants.
Humility and deference: Mephibosheth approaches David with respect, acknowledging David's kingly status and his own vulnerable position as a descendant of Saul.
The beginning of reconciliation: This is the initial encounter between David and Mephibosheth, setting the stage for David's subsequent act of grace and restoration.

The verse is important because it sets up a scenario where David could have easily chosen to eliminate a potential threat to his throne (Mephibosheth, being from Saul's line). Instead, it highlights David's commitment to his covenant with Jonathan and his intention to show mercy and kindness, even to those who might be seen as enemies.