Dumbledore doesn't just look into Harry's eyes, he uses a powerful magic called Occlumency on him. He's trying to understand what happened during Harry's vision, which is clouded by Voldemort's influence.
Here's why Dumbledore looks into Harry's eyes in that scene:
* To probe Harry's mind: Occlumency allows Dumbledore to peer into Harry's mind and see the memories he's blocked out. He's trying to understand what Voldemort did to Harry and why Harry is experiencing these visions.
* To teach Harry Occlumency: While the primary purpose is to understand what happened, it's also a teaching moment. Dumbledore is trying to teach Harry how to defend his mind against Voldemort's intrusions.
* To protect Harry: The visions are dangerous, and Dumbledore needs to know exactly what Voldemort is doing to Harry to help him protect himself.
The scene is important because it demonstrates the importance of the mind in the fight against Voldemort. It also highlights the close relationship between Dumbledore and Harry, who are both struggling with the darkness that threatens them.