Translation
On account of this, a woman is morally obligated to have authority on [her] head, on account of the angels.
Paraphrase
For this reason, a wife is morally obligated to show on her head a sign of respect to the authority that is over her; even the angels demonstrated the same practice.
WHY WERE THE ANGELS BROUGHT INTO THIS?
Paul was not thinking of all the angels, just of those that have this special thing they do to show the highest reverence in the presence of God. He was thinking of the Seraphim (mentioned only in Is 6:2 & 6), angels that have 6 wings. Two of those wings are used to cover their faces (because the glory of God is too great for them to behold, and thus, this is a recognition of God’s awesome splendor), and with two they cover their feet (because the feet often get soiled while going about one’s work and are therefore representative of weaknesses and limitations). It is our instinctive response to cover our weaknesses, even though God sees all things. This act is either an instinctive response on the part of these angels that is similar to our own, or it is an acknowledgement of weaknesses and limitations, or both.
Notice that these angels are covering up, not uncovering something, in God’s presence. Here Paul’s Jewishness finally comes out. Has he changed his mind about covering or not covering the head? No, Paul is simply saying that even certain angels do something special to show utmost reverence to God while in His presence. How that respect is shown does not matter as much as the fact that you are doing it.