For one thing, we know the joy it brings them to be saved, and thus share in their joy. This seems like a human sort of joy, however, and is therefore not able to fully account for the proper joy which a Christian has concerning another's salvation. What does scripture say? Paul often speaks of those who came to salvation through his ministry as his joy, for example, in 1 Thessalonians 2:19-20 "For what is our hope or joy or crown of boasting before our Lord Jesus at his coming? Is it not you? For you are our glory and joy." Christ, too, is said to give us praise and glory and honor.
In this you rejoice, though now for a little while, if necessary, you have been grieved by various trials, so that the tested genuineness of your faith—more precious than gold that perishes though it is tested by fire—may be found to result in praise and glory and honor at the revelation of Jesus Christ. Though you have not seen him, you love him. Though you do not now see him, you believe in him and rejoice with joy that is inexpressible and filled with glory, obtaining the outcome of your faith, the salvation of your souls. --1 Peter 1:6-9"The tested genuineness of your faith...may be found to result in praise and glory and honor at the revelation of Jesus Christ...obtaining the outcome of your faith, the salvation of your souls" This makes sense, in that it is his glory we have, we are united in him, so all that is his is ours. We express his nature in our very being, since it is Christ in us which gives us a heart, a being, of flesh. We can see it more clearly in Luke 15:4-10
“What man of you, having a hundred sheep, if he has lost one of them, does not leave the ninety-nine in the open country, and go after the one that is lost, until he finds it? And when he has found it, he lays it on his shoulders, rejoicing. And when he comes home, he calls together his friends and his neighbors, saying to them, ‘Rejoice with me, for I have found my sheep that was lost.’ Just so, I tell you, there will be more joy in heaven over one sinner who repents than over ninety-nine righteous persons who need no repentance.These are told in the context of Pharisees being annoyed that Jesus is hanging out with sinners as the pharisees say. So, Jesus seems to be saying that of course he is hanging out with sinners, they are the lost people, and there will be great rejoicing in Christ's finding them. Not only that there will be great rejoicing by the angels, but the stories portray Christ, the good shepherd, rejoicing himself when he finds what was lost. "Rejoice with me" he says. Why? So the question is also: why does Christ rejoice in uniting us to himself?
“Or what woman, having ten silver coins, if she loses one coin, does not light a lamp and sweep the house and seek diligently until she finds it? And when she has found it, she calls together her friends and neighbors, saying, ‘Rejoice with me, for I have found the coin that I had lost.’ Just so, I tell you, there is joy before the angels of God over one sinner who repents.”
It is entirely possible to say that our joy is found in Christ's joy, for just as Christ rejoices in uniting us to himself, so we rejoice in others being united to us in Christ. I would even argue that all of these joys are Christ's or our enjoyment of Christ. If we are saved, then we are crowned with the glory of God. "I have given them the glory that you gave me, that they may be one as we are one:" John 17:22. This would, then, provide an answer. Why do we and Christ find joy in others being joined to Christ? Because, in joining people to Christ, more of Christ is made visible. Christ's nature shows up in us when we are joined to him. Christ lives in us. In a person's salvation Christ is made visible, so we rejoice, just as we will rejoice so much more when Christ returns, that here is another image of Christ on this earth, another person self-consciously living for that very purpose for which they were made, that is, to the glory of God. Thus, the salvation of another brings joy to all who believe because it is to the glory of God, and because in them we will therefore see the glory of God work itself into the world where we can see it, even in their being changed into Christ's likeness. Thus, our joy in another's being saved is more than merely the change which is wrought in their life by the gospel, but the glory of God being made visible in that change. It is not the change as such, but what we see in and by the change: The grace of God to save sinners, the power of God to change dead lives into living lives, and the image of Christ made visible through them.