We’ll get to vestments in a minute, but first I think a little groundwork might be helpful.
Why is it that only an ordained Catholic priest can preside at Mass? The short answer is that Jesus set it up that way; you can read about it in various places in the New Testament (Luke 22:19 and John 20:21-23, to start with). Jesus gave the apostles the power to lead and teach, to celebrate Eucharist, and to forgive sins. The apostles passed that power, through the Sacrament of Holy Orders (ordination) on to the first bishops who passed it on down an unbroken line to your bishop today. Before long, the church grew and grew to the point where these bishops needed help, so priests and deacons were ordained to do specific jobs under the authority of their local bishop.
Jesus is the true and only high priest, and is the source of all priesthood. Through the Sacrament of Holy Orders, it is Christ himself who is now present as a priest performs a sacrament. For example, your confession is being spoken to Jesus and it is Jesus who forgives your sins. In another example consider this; any person could say the prayers of the Mass, but transubstantiation only takes place when an ordained priest says them. Ordination gives a priest the power and authority to act in the place and person of Christ himself!
Now on to today’s related topic.
Since your priest is very aware of the reality of Christ’s presence in his ministry, even a simple action like getting dressed for Mass is an opportunity for prayer. Before he begins putting on vestments, he washes his hands and prays that Jesus will give him the grace of purity. He puts on his alb and prays that through the grace of his Baptism that he will be wholly devoted to Jesus. While tying on his cincture (the cord tied around his waist) he prays that Jesus will strengthen him with the virtue of chastity. The stole is a signal that he’s “on duty” and ready to officiate at a Sacrament. While putting it on, he prays that Jesus will make him worthy of this awesome job. And finally, he puts on a chasuble which covers everything else. This is a symbol of charity (love) and, as he puts it on, your priest prays that all he does is covered by the love of Jesus. A very common version of this prayer is this Vesting Prayer. You can learn more about the meaning of the vestment pieces as you read through it, and since many of them are included with your priest paper doll, you can read while your child dresses Father for Mass.
And for grown-ups who want to learn more, I recommend this short piece on liturgical vestment from the Vatican’s web site.