This verse is a pivotal point in the story of Amnon's plot to rape his half-sister Tamar. Let's break down what it means:

"So Amnon lay down and faked being sick.": Amnon, driven by his lust for Tamar, is pretending to be ill. This is the first step in his deceitful plan. He's using illness as a pretext to manipulate the situation.

"When the king came to see him,": This shows the king (David, their father) cares for his children and is concerned about Amnon's supposed illness. It also highlights Amnon's audacity and his willingness to exploit his father's affection.

"Amnon said to the king, “Please let my sister Tamar come, and make me a couple of cakes in my sight, that I may eat from her hand.”": This is the core of Amnon's scheme. He asks David to send Tamar to him to bake cakes in his presence. This request is deliberately designed to:

Get Tamar alone with him: This is his primary objective. He needs to isolate Tamar to carry out his plan.
Make his request seem innocent: Asking for a family member to prepare a simple meal appears harmless. It masks his true intentions, deceiving both David and Tamar.
Place Tamar in a vulnerable position: By asking her to serve him in his room, he creates a situation where he has power and control over her.
Exploit the cultural norms of hospitality: In that time, it was common for family members, especially women, to serve and care for sick relatives. Amnon is using this expectation to his advantage.

In essence, this verse reveals Amnon's cunning and manipulative nature. He is lying to his father, exploiting his sister's willingness to care for him, and setting a trap to fulfill his lustful desires. It sets the stage for the tragic and violent events that follow. It is important to note that his request is unusual, as generally servants would prepare food, not a princess like Tamar. This should have raised suspicion, but David is apparently unware of his son's true motives.