Why did he say this to him?
This is a great question. The answer involves a bit of knowledge about American culture and law. "You have the right to remain silent" is the most well-known line from the Miranda warning, also known as the Miranda rights. The Miranda rights are recited by police officers when arresting criminal suspects, in order to inform them of their right to remain silent. In this context, it is likely that SpiderMan has quoted this line to hint that he is aware of the crimes that Bucky has committed. Ex. You are under arrest! You have the right to remain silent.