When a royal wife enters the king's private room, the meeting usually reflects their personal relationship rather than formal court rules. However, Bathsheba approaches King David with a surprising level of ceremony, showing right away that her visit is not a normal one, but a matter of great urgency.
She bows her head downward and completely lowers herself to the ground [מצודת ציון]. This display of royal manners is highly unusual for a wife visiting her husband's private space [מלבי״ם]. Her arrival alone catches his attention, either because she does not normally visit him on her own, or because she enters the room without asking for permission first [רד״ק, מצודת דוד].
By combining this unexpected visit with such formal bowing, she makes it clear that she has come to ask for something special [מלבי״ם]. King David immediately sees that this is a serious situation. Watching her unusual behavior, he understands that she brings an important and necessary matter [מלבי״ם, רד״ק]. Because of the way she acts, he asks her what has happened and what she needs, inviting her to share her request [ביאור שטיינזלץ].