When at his Last Supper, Jesus said: "This is my body", what he held in his hands still had all the appearances of bread: these "accidents" remained unchanged. However, the Roman Catholic Church believes that, when Jesus made that declaration, the underlying reality (the "substance") of the bread was converted to that of his body. In other words, it actually was his body, while all the appearances open to the senses or to scientific investigation were still those of bread, exactly as before.
The Catholic Church holds that the same change of the substance of the bread and of the wine occurs at the consecration of the Eucharist
So, how can the Eucharist change from bread to body if it is not measurably altered in any way? How is it different from saying that it doesn't change at all? Is it an allegory? If not, how does it change in practical terms?
Transubstantiation relies on the Aristotelian distinction between essence and accidents. From the linked Wikipedia article:
Catholic theologians such as Thomas Aquinas have employed the
Aristotelian concepts of substance and accident in articulating the
theology of the Eucharist, particularly the transubstantiation of
bread and wine into body and blood. According to this tradition, the
accidents of the bread and wine do not change, but their substances
change from bread and wine to the Body and Blood of Christ.
The "measurable" properties you are referring to (taste, feel, etc.) are accidental properties, not the substance of the bread/wine.
The Real Presence: that the true body of Christ the Lord, the same that was born of the Virgin, and is now seated at the right hand of the Father in heaven, is contained in this Sacrament.
Transubstantion itself: that, however repugnant it may appear to the senses, none of the substance of the elements [bread or wine] remains in the Sacrament. (This is against impanation, which believes the substance of the bread and wine exists alongside Christ in the Eucharist.)
The Mystery of the Accidents without a Subject may be deduced from the two preceding. Although the words of consecration themselves clearly express it, is that the accidents which present themselves to the eyes or other senses exist in a wonderful and ineffable manner without a subject. All the accidents of bread and wine we can see, but they inhere in no substance, and exist independently of any; for the substance of the bread and wine is so changed into the body and blood of our Lord that they altogether cease to be the substance of bread and wine.
Practically, it changes because the Eucharist is Jesus' Body, Blood, Soul and Divinity made present before you.
Previous to consecration it required no faith to believe that bread and wine were bread and wine. After consecration it requires absolute faith to see Our Lord before us (although the reality of the Real Presence is there whether one believes it or not).
Transubstantiation is a miracle and like most miracles your freewill and liberty are sublimely preserved even when it happens in front of your eyes.