Why say prayers for priests? Because, as St. John Vianney, the patron saint of parish priests, once said “After God, the priest is everything.” (He also once referred to the priest “the steward of the good God, the distributor of His wealth.”)
In administering the sacraments (which only they can do) and serving us, priests act, as we say, in persona Christi (that is to say, in the person of Christ), with our Lord working through their presence.
(As a quick reminder, the sacraments are seven outward signs Christ instituted to give us grace, a supernatural gift from God to enable us to grow in goodness and holiness. These are: Baptism, Confirmation, the Eucharist, Penance, the Anointing of the Sick, Holy Orders, and Matrimony. The links given here refer to Chapters from the Catechism of the Catholic Church where you can read more about each one.)
As Father John Hardon, S.J., once said, “praying and offering God sacrifices for the priesthood are indispensably important,” because “there is no Catholic Church without the priesthood.”