Here are 8 reasons why Jesus is better than angels:

he has the superior name - that is title, character, essence (v4)
He is the Son - that is the King (v5)
God is his father in a unique way (v5b)
God commands angels to worship him. The lesser worship the greater (v6)
Despite their mighty nature and power, angels are still servants while the Son is God and King (v7-8)
The Son, not angels, made Earth and Heaven (v10)
The Son will eternally outlast the current Heavens and Earth (v11)
The Son is exalted to God's right hand until all his enemies are conquered (v13)

Pretty strong case. However the important thing is not that Jesus is superior to angels, it is why the writer needs to prove it. The main reason is to show that the message of God communicated by the Son is greater than the message of the Mosaic Law, which was safeguarded by angels.

I imagine people saying "how can you possibly claim that this good news of Jesus supercedes the Law, when that was attended by such heavenly power and safeguards? Surely the Law was the ultimate revelation of God, and of how to know God? Hebrews blows that out of the water. The good news of Jesus overwhelmingly surpasses the Law, by the extent that Jesus' identity is superior to the angels' identity. If you have a brief look forward to chapter 3 you will see that the culmination of the Jesus-superior-to-angels argument is that Jesus is greater than Moses. That's the point.

A couple more reasons why the writer makes such a lengthy case in the next post, but the conclusion here is that we don't trust our performance of the Mosaic Law to get right with God, we trust Jesus. This was written to people who banked all their hopes on Law-keeping-for-righteousness. It failed them miserably.

There is no being right with God through Law-keeping. We are bankrupt, nobody keeps it. That's the single big problem for the human race that is at the very heart of the good news - nobody lives up to the standard of the glory of God, and therefore all get judged and condemned. All are under his curse. But, says Hebrews, there is a new message of complete rescue and real righteousness that comes in Jesus Christ. In fact later he will say that anyone who belongs to Jesus is declared "perfect for ever" because of Jesus' sacrifice for them (10:14).

If you are reading this and you aren't a Christian yet, you can have that verdict proclaimed over your life by God - perfect for ever. Just by trusting the Son, and making him Lord over your life. Stop working at being good enough - you will fail perpetually - and ask him to do it for you. He will. And its brilliant.