What does Matthew 8:17 mean?

The previous verse describes an evening of healing and casting out demons at Peter's house in Capernaum (Matthew 8:16). Jesus welcomed all who needed help in either of these ways and took care of their issue in a moment.

Now Matthew points to this evening, along with all the other healing Jesus did during His ministry on earth, as the fulfilment of a specific prophecy found in Isaiah 53:4. Matthew quotes that verse in this way, "He took our illnesses and bore our diseases." Matthew's primary audience for this gospel is the nation of Israel. Points such as this are meant to tell Jewish people, in particular, that Jesus truly is the Messiah. Other portions of Isaiah chapter 53 describe the suffering Messiah would experience as part of His ministry (Isaiah 53:5; 1 Peter 2:24; Mark 10:33).

In His eager willingness to heal all in Israel who were sick and to cast out their demons, Jesus demonstrated the compassionate nature of God and showed that He came to serve Israel, even as her King.