I don’t think Paul’s intent was to teach about the end of times or our sitting in judgement of angels. The whole chapter is about the Corinthian’s use of pagan courts of law, which were based on pagan ideas of morality and punishment instead of the Christian ideals of mercy and justice. Paul is using a rabbinical teaching technique where he is asking a series of rhetorical questions based on some teaching in the Old Testament and following each with a question about his current audience’s incorrect behavior in light of that teaching. The formula is…
Do you not know…? Then why do you do…?
In asking the question in the negative (do you not know?), he is almost insulting his audience, as if he was aksing them, “Didn’t you learn this already when you were little kids? Do we have to go over this again?” It’s brilliant, and very rabbinical.
***Do you not know that the holy ones will judge the world? If the world is to be judged by you, are you unqualified for the lowest law courts? ***
My poor attempt at a translation would be: If you are to sit in judgement of the world, then why do you feel unqualified to set up your own courts so that you don’t have to use pagan courts?
***Do you not know that we will judge angels? Then why not everyday matters? ***
Again, the same sentiment is expressed. A focus on judging the angels in that passage, while interesting, misses Paul’s point, which is that the secular, pagan and even rabbinical courts of the Jews were not based on Christian teaching about justice and mercy in light of every Christian’s status as members in Christ’s body.
***Do you not know that your bodies are members of Christ? Shall I then take Christ’s members and make them the members of a prostitute? Of course not! *
(1 Corinthians 6:15)
If the Corinthians united themselves to Pagan justice, then what was to stop them, like Israel in the Old Testament, from uniting themselves to pagan practices like boy prostitution, robbery, and the rest of the practices listed in verse 9.
Paul was a Pharisee and an Old Testament scholar who studied under the famouns Rabbi Gamaliel. He far outpaced his fellow students and was something of a prodigy. Even Peter has trouble understanding him at times.
***…our beloved brother Paul, according to the wisdom given to him, also wrote to you, speaking of these things as he does in all his letters. In them there are some things hard to understand… *
(2 Peter 3:16)
If Peter found Paul difficult to understand, then I’m in good company.

Personally, even if I never understand what the reference to judging angels is about, it doesn’t matter to me, because it is not central to what Paul is trying to teach. To answer the original question, my opinion is that sitting in judgement of the angels is a reference to Daniel 7, where we read
“Then the kingship and dominion and majesty of all the kingdoms under the heavens shall be given to the holy people of the Most High, Whose kingdom shall be everlasting: all dominions shall serve and obey him.” (Daniel 7:27). But that’s just my opintion. The passage does speak to the idea of the unity in Christ that his members havewith him as in the commentary cited by Little One0307. The entire book of Daniel is something every Catholic should be intimately familiar with and Both Paul and Jesus reference the book of Daniel extensively.
But again, I think judging angels is not Paul’s central theme, that it was simply a teaching technique used to get his point across. I hope I did not veer off topic but will invoke the “Other comments” clause established in the original post.
PAX
-Tim-