I get two things from this passage.
1. The angels will go through some sort of judgment at the end time.
2. We (ie faithful disciples or some subset of the faithful disciples of Jesus) will be there for this judgment.
I do not believe that we will do the actual judging. I believe that God will do the actual judging, but I believe that we will be there as witness, perhaps even giving assent to God’s judgments.
I will have to say that this seems a bit surprising to me (and maybe to you as well). If you had asked me, and it was not stated in the Bible, I would have assumed it might be the other way around. I would have assumed that the angels are above humans in the hierarchy scale in heaven. But I would be wrong. In fact, Hebrews 1:5-14 and Hebrews 2:5-9 agrees with 1 Cor 6 that, in heaven, in the final times, humans will occupy a place at least in some way superior to that of angels. In this position we will be involved in some unstated way in their judgment at the end time. Exactly what our role will be I am not sure and I choose not to speculate.
But this gets us to the point that Paul is trying to make. His purpose is not to tell us about how angels will be judged. I do not think that is all that important to Paul. The point in the context of 1 Cor 6:3 is that the disciples in Corinth ought not to be placing their disputes before worldly judges. People in the church should be judging conflicts between people in the church, not sending them off to the world. So, let me advise you to not get all that worried about the side issue, which is what it means that we will judge angels, but that you should be concerned about what God is concerned about, which is unity in the church.