The salvation that Peter is referring to is described previously. He was encouraging us that even though our salvation is mysterious it is very valuable, and that the inexpressible joy that we feel is not unusual or a cause for concern. Don't be uncomfortable. Grace and peace.
To encourage us even more, Peter tells us that even the prophets (he implicitly assumes his readers are familiar with the prophets, so I guess they're definitely Jewish, as guessed in the first post in this series) searched for understanding about our salvation.
They wanted to know who the Christ would be and when he would arrive. I guess they initially thought they were doing this to satisfy their own curiosity because it had to be revealed to them that this was in fact a service to the saints that would come after them.
Interestingly, it says that it was the Spirit who predicted Christ's sufferings and the following glories. I would not have expected that the Spirit of God needed to do any predicting about Christ's sufferings or glories. Are you even able to predict something if you're omnipotent?
Back to the text, Peters says that things have been announced to us via those who preached the good news to us by the Hold Spirit, who was sent from heaven.
Preaching by the power or enabling of the Holy Spirit has revealed things to us that angels long1 to know. We don't know everything, and we still have questions, but some of our questions are answered, some of them inexpressibly, and some of the things we find mysterious or uncomfortable are supposed to be so until our salvation is complete. Grace and peace. Hang in there, you're doing OK.
Long, or longed? I would have expected it to be "longed to know", but its not. Does chronology get messed up from our perspective if the other party is outside of time? ↩