This verse is part of a conversation between Bathsheba, the mother of King Solomon, and Adonijah, Solomon's half-brother. Here's a breakdown of its meaning and context:

Context: Adonijah had previously attempted to usurp the throne from Solomon. Bathsheba, aware of the tension and Solomon's potential suspicion of Adonijah, visits Solomon. Adonijah asks Bathsheba to intercede on his behalf with Solomon.

"Now I ask one petition of you. Don’t deny me." Adonijah is appealing to Bathsheba for a favor and emphasizing the importance of her agreeing to help him. He's setting up his request as something he believes is significant to him and that he needs her to grant.

"She said to him, 'Say on.'" This is Bathsheba's response. It means "Tell me what you want," or "Go ahead and ask." She's agreeing to listen to his request. It does not mean she agrees to grant it, only that she's willing to hear him out.

In essence, the verse is a setup for Adonijah's request. He's making a plea to Bathsheba, and she's giving him permission to state what he desires.

Why is this significant?

Power Dynamics: The interaction reveals the power dynamics at play. Adonijah knows he needs Bathsheba's influence with Solomon.
Foreshadowing: It creates suspense. The reader knows that Adonijah's request could have major consequences, especially given his past actions and Solomon's likely distrust of him.

The story unfolds with Adonijah asking Bathsheba to ask Solomon to give him Abishag the Shunammite as his wife. This request is seen as another attempt to claim the throne, and it leads to Adonijah's execution by Solomon. So this verse, while seemingly innocuous, is a critical moment in a power struggle.