In Hebrews 5:1-10, the writer to the Hebrews began explaining the theme of the Great High Priestly nature of Jesus - but when he got to Hebrews 5:11, he paused in his line of thinking because he came to realise that there was something not quite right in the hearts and the minds of those he was writing to. So from Hebrews 5:12, right through until the end of chapter 6, the writer has been bringing a strong exhortation for his readers to press forward in spiritually maturity. But now that he has addressed this heart issue, the writer now picks up in Hebrews 7:1, where he left off in Hebrews 5:11, to continue talking about Jesus as our Great High Priest.
In chapter 7, the author’s main aim is to firstly demonstrate that Jesus was in fact a legitimate High Priest. The Jews in that day knew that strictly speaking, Jesus did not come from the High Priestly line of Aaron and the Levites, so some may have had questions about the legitimacy of Jesus’ High Priestly function. But what the writer does, is that he demonstrates from Scripture that Jesus actually comes from another priesthood - a greater priesthood. So as the writer explains this, what we get is a wonderful picture of the ministry of Jesus for His people both back then as well as today!