It just occured to me that this happened just after He breathed the Spirit on them. I wonder if it was to do with that and Jesus empowering them fof their task ahead?
Well > the Holy Spirit would make them able to have compassion for people who need forgiveness, so the forgiving can be done kindly and generously . . . "even as God" (Ephesians 4:31-32).
So, they would be given authority, yes, and more . . . the way of personal loving needed in our forgiving. The keys to Christ's kingdom are keys to have authority and power, yes, but even more-so to have family tenderly caring and sharing love which is in Jesus alone . . . in His Holy Spirit.
Forgiving is not only a legal process, then, of keeping a person's record clean.
Forgiving can be treated like a thing of authority and power . . . of having special status over others. But our Apostle Peter says how leaders need to lead >
"nor as being lords over those entrusted to you, but being examples to the flock." (1 Peter 5:3)
How they forgive can be our example. And our Apostle Paul says Jesus on the cross is our example to follow >
"And walk in love, as Christ also has loved us and given Himself for us, an offering and a sacrifice to God for a sweet-smelling aroma." (Ephesians 5:2)
So, while Jesus was getting forgiveness for us, He was being sweet-smelling about it. We need to prepare in prayer so we are able to sweetly and kindly and generously forgive.
On the cross our example Jesus prayed > "Father, forgive them, for they do not know what they do." (in Luke 23:34) Even while, I would say, those people could care less about being forgiving, already Jesus in His heart was forgiving them and so He prayed. And then, a few verses later . . . we see how one of those criminals hanging next to Christ asked Jesus to remember Him. Possibly, then, I would say, that prayer worked to get that man forgiven and changed so he trusted in Jesus for salvation and became Jesus Christ's own brother; he even stood up for Jesus.
And we are directed by our Apostle Paul to so walk in love, as Christ loved us.
And yes the Holy Spirit in us has us lovingly forgiving like this, including with prayer for people, not against them. We have been ministered this "Spirit of adoption" (Romans 8:15) who makes us likewise lovingly adoptive and generously forgiving like Jesus.
Jesus and His "examples to the flock" (1 Peter 5:3) are our example of this which is for all of us.