Jesus often used strong language when those he was speaking to were clearly in error! See Matthew 23:27 for example:

“Woe to you, teachers of the law and Pharisees, you hypocrites! You are like whitewashed tombs, which look beautiful on the outside but on the inside are full of dead men’s bones and everything unclean." (NIV)

Or in John 8:44 Jesus rebukes the Pharisees in the following terms:

You belong to your father, the devil, and you want to carry out your father’s desire. He was a murderer from the beginning, not holding to the truth, for there is no truth in him. When he lies, he speaks his native language, for he is a liar and the father of lies. John 8:44 (NIV)

But the rebuke to Peter in Mt 16:22 seems especially strong because He rebukes him as if Satan himself is standing there.
It is forceful language and is designed to show Peter that his thoughts are directly opposed to Jesus' purpose and mission so that they could only have been inspired by Satan. In fact any thought, attitude or action that is in conflict with Jesus mission is inspired by the evil one!

In the context of the passage Jesus has announced that the climax of his mission on earth is to be his death and resurrection (Matthew 16: 21). Peter didn’t like what he heard! He was probably thinking that Jesus would enter Jerusalem as a strong political figure who would defeat the Romans and take the place of the king of Israel physically. Perhaps he was hoping that he might have an important position in Jesus’ government!

So to Peter the idea of Jesus dying was unthinkable. He hadn’t realised that the path to Jesus’ glory was through the suffering of death upon the cross. Peter’s thoughts were self-centred. He hadn’t realised that Jesus mission was bigger than he could ever have imagined.

The connection with Satan goes back to Jesus’ temptation in the wilderness where Satan was tempting Jesus to take the Kingdom without going to the cross! (See Matthew 4:1-11). In other words to have power and glory but without the suffering of His sacrifice. That must have been a very attractive temptation for Jesus because the anticipation of His suffering later caused Him to sweat drops like blood in the garden on the night of His betrayal (Luke 22:44).

Peter was meant to be a disciple – one who was learning from the teacher – to rebuke Jesus as he did with this same temptation Jesus had faced from Satan must have been a very low point in the relationship between Peter and Jesus!

Yet in Luke 22:31-32 before Peter denies Jesus three times Jesus says to him:

“Simon, Simon, Satan has asked to sift you as wheat. But I have prayed for you, Simon, that your faith may not fail. And when you have turned back, strengthen your brothers.” (NIV)

Unknown to Peter he had been specifically targeted by Satan, but Jesus had countered this by praying for him. Peter did go on to learn from his mistakes. In fact after Jesus reinstated Peter (see John 21:15-17) he went on to become a pillar of the church (see Galatians 2:9) just as Christ said: “you are Peter, and on this rock I will build my church” Matthew 16:18 (NIV).