He put the wristwatch under Hassan's mattress
Amir frames Hassan for stealing money and a watch, knowing that Hassan will take the blame to protect him. Amir stands by and does nothing to stop it, thus indirectly getting both Ali and Hassan in trouble.
Hassan and Ali give Amir a leather-bound notebook for his birthday.
Amir put his watch and some money in Hassan's mattress. Hassan found these items and told Ali about how he got raped while Amir hid and watched, so they left Kabul.
Hassan is Amir's half brother. Hassan's father, Ali, was like a brother to Baba so he lived with Baba. However, Baba impreganated Ali's wife and had Hassan who became good friends with Amir.
Ali and his son Hassan left Baba's household in "The Kite Runner" because they were forced to leave due to the dishonor of Hassan's biological father, which created tension and conflict in the household. Amir frames Hassan for theft to get rid of him, leading to Ali and Hassan's departure as a result of the betrayal and injustice they faced.
Actually, they are half brothers. Amir had found out that when Rahim Khan had hinted that Ali was not Hassan's father but Baba was (which explains why Baba had a closer relationship to Hassan and Amir).
Ali tells Amir that Hassan mostly runs errands and does chores for him now, as he is getting older and can no longer do physical work like he used to.
Baba shocks Amir by letting Hassan and Ali stay at the house, as Amir had put the wristwatch and Afghani bills under Hassan's mattress so that when Baba did find out about Hassan's stealing (a trait that Baba once told Amir he would never forgive in a person), Baba would want to push them out. Instead of Baba kicking them out, he forgave Hassan for "stealing" (he only admits stealing from Amir to protect him), and begs them to stay. Hassan and Ali leave the next morning and Baba weeps when they leave.
Amir is selfish and Hassan is humble and loyal to Amir. Hassan is a Hazara, Amir is a pashtun. Amir is granted to right to get an education, Hassan is not.
Amir remembers how his father, Baba, risked his reputation and safety to save Hassan from harm at the kite-fighting tournament, displaying his loyalty to Hassan despite the societal divide. Amir recalls how Baba treated Hassan with affection and fairness, providing for him, defending him, and standing up for him, which shows his unwavering loyalty towards Hassan even though Amir didn't fully understand it at the time.
Hassan is revealed to be Baba's son, making him Amir's half-brother. Baba had an affair with Hassan's mother, resulting in Hassan's birth. This revelation adds a new layer to the complex relationships in "The Kite Runner."
Hassan doesn't tell the truth about Amir to protect their relationship and honor. He values their friendship and doesn't want to break the trust between them, even if it means taking the blame for something he didn't do. Hassan's loyalty and selflessness are stronger than his desire to seek justice for himself.
Amir asked Hassan to leave the neighborhood and move away from the household, as it would be easier for Hassan to live a peaceful life without the burden of their complicated relationship and the secrets that were kept from him.