The Catholic Church believes that this has been the tradition since "the beginning" of the Church.
Firstly, according to the Church, Christ makes it very clear in John 6:41-58 that the literal, physical consumption of His body is ultimately intended. The phrasing used in this passage, particularly in the Greek, is very unambiguous. As compared to other passages, wherein Christ clarifies figurative language, in John 6, when pressed to clarify, he intensifies the language, using a more animalistic and "gory" verb to denote the eating/gnawing of His own flesh.
Secondly, the Church sees the Last Supper as the formal establishment of the Eucharist. Christ says a blessing, the exact words of which are not included in the Gospel, but were undoubtedly remembered by the 12 and/or were simply commonplace blessings. He declares the bread and wine to be His body and blood. He distributes it. And He commands the disciples to repeat what He's done (by calling it a covenant in Matthew and Mark, and by also explicitly commanding it in Luke). Especially in the context of John 6:41-58, it's unambiguous in Catholic belief.
14 When the hour came, he took his place at table with the
apostles. 15 He said to them, “I have eagerly desired to
eat this Passover with you before I suffer, 16 for, I tell
you, I shall not eat it [again] until there is fulfillment in the
kingdom of God.” 17 Then he took a cup, gave thanks, and
said, “Take this and share it among yourselves; 18 for I
tell you [that] from this time on I shall not drink of the fruit of
the vine until the kingdom of God comes.” 19 Then he took
the bread, said the blessing, broke it, and gave it to them, saying,
“This is my body, which will be given for you; do this in memory of
me.” 20 And likewise the cup after they had eaten, saying,
“This cup is the new covenant in my blood, which will be shed for you. (Luke 22:14-20)
Speculating into the Church rationale a little: It may be worth mentioning that in the general command to eat His flesh and drink His blood is given to a crowd / the general faithful. In this Last Supper discourses, however, Jesus is with "the twelve," with whom He often gives special commands and teachings not given to the general faithful. Thus, the Church may be concluding from this the command to consecrate the bread and wine is given only to the Apostles and their successors.
So, the framework is set in the Gospel. Christ establishes a set of procedures for the disciples to follow, by which they are to consume His body and blood (of "the new and everlasting covenant"). Historically, the practice is referred to in the New Testament and early Church writings.
In the New Testament, for instance, 1st Corinthians makes it clear that the Eucharist was common practice. It's also fairly clear that practice thereof followed a particular pattern, and that reception thereof was to be taken very seriously.
23 For I have received of the Lord that which also I
delivered unto you, that the Lord Jesus, the same night in which he
was betrayed, took bread, 24 and giving thanks, broke and
said: Take and eat: This is my body, which shall be delivered for you.
This do for the commemoration of me. 25 In like manner also
the chalice, after he had supped, saying: This chalice is the new
testament in my blood. This do, as often as you shall drink, for the
commemoration of me. 26 For as often as you shall eat this
bread and drink the chalice, you shall show the death of the Lord,
until he come. 27 Therefore, whosoever shall eat this
bread, or drink the chalice of the Lord unworthily, shall be guilty of
the body and of the blood of the Lord. 28 But let a man
prove himself: and so let him eat of that bread and drink of the
chalice. 29 For he that eats and drinks unworthily eats and
drinks judgment to himself, not discerning the body of the Lord. (1
Outside the New Testament, the Didache, among copious other writings, confirms the practice.
Now concerning the Thanksgiving (Eucharist), thus give thanks. First,
concerning the cup: We thank you, our Father, for the holy vine of
David Your servant, which You made known to us through Jesus Your
Servant; to You be the glory for ever. And concerning the broken
bread: We thank You, our Father, for the life and knowledge which You
made known to us through Jesus Your Servant; to You be the glory for
ever. Even as this broken bread was scattered over the hills, and was
gathered together and became one, so let Your Church be gathered
together from the ends of the earth into Your kingdom; for Yours is
the glory and the power through Jesus Christ for ever. But let no one
eat or drink of your Thanksgiving (Eucharist), but they who have been
baptized into the name of the Lord; for concerning this also the Lord
has said, Give not that which is holy to the dogs. (Didache
Summa Theologica, question 75, Article 2 further confirms some historical tradition of the belief in transubstantiation and that it through the act of consecration that the Lord makes Himself physically present in the species of bread and wine.
Beyond that, I'd suggest taking a look at the Catholic Encyclopedia article on the Real Presence. In particular, see the Proof from Tradition section.
In terms of why the laity cannot yield the effect of transubstantiation, I don't have an extensive enough knowledge to pin that question down precisely. But, the general tone of the NT letters and the Didache is that ordained bishops and deacons (Apostles) and prophets retain special rights to speak and perform in Church gatherings, Eucharistic celebrations, and so forth. The distinction is made in the Catholic Church between Apostles and Disciples -- anyone and everyone can be a disciple, but not all are Apostles, if I'm not mistaken.
From a more theological standpoint, bear in mind that Catholic theology is just generally sacramental, wherein specific things, individuals, group members, or entire groups represent or reveal spiritual realities. As such, it detracts from the sacrament if the symbolism is lost in some way, such as if the priest's Church-given, sacramental, iconic role is confounded or infringed on. The priest is assigned, in ordination, as a sacramental icon of Christ. If anyone is allowed to step in and out of this iconic role at any given time, the sacramental nature is largely lost.
To illustrate that point of Catholic understanding, consider an umpire. Two teams and a host of onlookers assemble and agree, more or less, to operate under the ruling of an "ordained" umpire, whose job it is to represent [and enforce] the rules of the game. If anyone were free to step in at any time to make a call, the rules become blurry -- they have no unbiased body or icon on the field anymore. And if an this game, wherein anyone can step in to make a call, were to go on for too long, the rules would change in-play and the game would devolve in to chaos.
Now, if we can imagine the loss of integrity and the rise of chaos that naturally takes place in petty human affairs by blurring iconic lines, imagine, says Catholic theology, how much more dangerous it is to blur the lines around sacramental icons (icons representing spiritual realities). We lose sight of truth very quickly. And in fact, any authority or "real" effect is immediately lost once we desecrate an iconic role -- even in petty human affairs! It makes a huge difference whether someone in the stands yells, "you're out" or the game's designated umpire yells, "you're out!" Much greater is the difference, says Catholic theology, in sacramental affairs.