Jesus referred to himself as the Son of God to demonstrate his divinity and the Son of Man to highlight his humanity.
Throughout his time on Earth, Jesus referred to God as his Father. By doing this he made it clear that he was more than just a human being. He also demonstrated this through miracles such as walking on water.
As his disciples got to know him, they came to understand who Jesus was. When Jesus asked some of them who they thought he was, one of them replied: 'You are the Christ, the Son of the living God' (Matthew 16:16).
Jesus also called himself the Son of Man. He said: 'The Son of Man did not come to be served, but to serve, and to give his life as a ransom for many' (Matthew 20:28).
His listeners at the time would have understood this as Jesus showing himself to be like them, in that he felt human emotions. He ate and drank. He slept. He formed relationships with the people around him. In basic terms, Son of Man means ‘human being’.
By calling himself the Son of God and the Son of Man, then demonstrating his spiritual authority through his teachings and miracles, Jesus was telling his followers something extraordinary about himself: yes, he is an ordinary human being, but he also possesses the sovereign power of God.