From the point of view of Catholicism, the first part of this question might be, "Why does anyone at all need to consecrate the Eucharist?" After all, if no one needs to consecrate it, then a fortiori it is not necessary for a priest to consecrate it.
As usual, I went to the Summa Theologica to check out what the Church's greatest theologian had to say. I wasn't disappointed. The Third Part of the Summa, Question 78, Article 1 discusses the wording by which the consecration takes place. As part of his discussion of the question, Aquinas states that "the form of this sacrament is pronounced as if Christ were speaking in person" (emphasis added); that is, by the use of the words "This is my body ... this is my blood" (cf. Luke 22:19–20, Matthew 26:26–28, 1 Corinthians 10:16).
The wording becomes important in the Third Part of the Summa, Question 82, Article 1, where Aquinas asks "Whether the consecration of this sacrament [of the Eucharist] belongs to a priest alone?"
Aquinas considers a number of objections. Most important, he asks whether it might be the case that any believer could consecrate the Eucharist. There are a couple of reasons why people might believe this is the case. They might be thinking that since the words themselves are the form of the consecration, therefore anyone who can say the words can consecrate the Eucharist. Or they might be thinking that lay people can consecrate the Eucharist by virtue of their participation in "the priesthood of all believers", by which
the person baptized is incorporated into the Church, the Body of Christ, and made a sharer in the priesthood of Christ.
(Catechism of the Catholic Church, paragraph 1279)
But Aquinas concludes that this is not sufficient to allow the believer to consecrate the Eucharist. He says,
Such is the dignity of this sacrament that it is performed only as in the person of Christ. Now whoever performs any act in another's stead, must do so by the power bestowed by such a one. But as the power of receiving this sacrament is conceded by Christ to the baptized person, so likewise the power of consecrating this sacrament on Christ's behalf is bestowed upon the priest at his ordination: for thereby he is put upon a level with them to whom the Lord said (Lk. 22:19): "Do this for a commemoration of Me." Therefore, it must be said that it belongs to priests to accomplish this sacrament.
In other words, what's specifically important about those words in particular are that Christ Himself said them. Thus, only one who is empowered to act as Christ will in fact consecrate the Eucharist by saying those words.
Now the Catholic Church does in fact believe that those in the higher ranks of Holy Orders (priests and bishops, but not deacons) are empowered to act as Christ:
The ministerial priest, by the sacred power he enjoys, teaches and rules the priestly people; acting in the person of Christ, he makes present the Eucharistic sacrifice, and offers it to God in the name of all the people.
(Lumen Gentium, section 10)
The encyclical letter Mediator Dei, written by Pope Pius XII in 1947, points out John 20:21 and Luke 10:16 in particular as instances where Christ sends out the apostles with his own power and authority; and the Catechism points out the institution of the seventy elders with the spirit that was in Moses (Numbers 11:24–25), in a prayer that is still said at the consecration of priests:
Lord, holy Father,...
when you had appointed high priests to rule your people,
you chose other men next to them in rank and dignity
to be with them and to help them in their task....
you extended the spirit of Moses to seventy wise men....
You shared among the sons of Aaron
the fullness of their father’s power.
Thus, the Church believes that as God has from early times, he pours out his Spirit on certain men, giving them the power to act as his delegates and ministers in a special way. It is by this outpouring and this grace of the Father that a priest, and only a priest, has the authority to speak in the person of Christ, and thus only a priest has the ability to consecrate the Eucharist.