Verse 13 derives from Psalm 110:1, a psalm considered Messianic even by Jewish scholars. We can think of this quotation as a reverberation of verse 4, in that we need to consider it in the same way as what is written there of the eternal God, Jesus Christ: “. . . having become so much better than the angels, as He has by inheritance obtained a more excellent name than they.”
The interesting phrase here is “having become.” But how can the eternal, constant God—eternally superior—change or “become so much better”? Of course, the Son was always eternally superior to the angels. That, however, is not what is in play here. The writer refers to what the Son did in becoming human and putting away the sins of men. Because He had paid the penalty for sins, He could sit down on God's throne in the place of the highest honor, and from this standpoint, He is deemed greater than any angel. No angel has ever come close to achieving that magnificent triumph.
Regarding angels, Hebrews 1:14 asks rhetorically, “Are they not all ministering spirits sent forth to minister for those who will inherit salvation?” This conclusion to the chapter makes a definitive statement about angels as compared to Jesus Christ. They have a place of dignity and honor in the Kingdom of God, but they are servants. The term “all” applies to every one of the angels without distinction. Despite their uncommon excellence in many areas, they are set aside to serve, especially the saved among mankind.