Yes, Catholics do believe that the bread and wine are transformed into the body and blood. Sort of. The "sort of" is because the technicalities of it go back to Aristotelian philosophy, which greatly influenced Thomas Aquinas, who is still in many ways the preeminent theologian of the Catholic Church.
Before we get to Aquinas, though, let's look at the modern Catechism of the Catholic Church. This is a teaching document, not primarily a theological discussion, so it doesn't go into technical specifics the way we'll see that Aquinas did. But we see (numbers are paragraph numbers):
1333 At the heart of the Eucharistic celebration are the bread and wine that, by the words of Christ and the invocation of the Holy Spirit, become Christ’s Body and Blood. Faithful to the Lord’s command the Church continues to do, in his memory and until his glorious return, what he did on the eve of his Passion: "He took bread...." "He took the cup filled with wine...." The signs of bread and wine become, in a way surpassing understanding, the Body and Blood of Christ; they continue also to signify the goodness of creation.
1353 In the epiclesis, the Church asks the Father to send his Holy Spirit (or the power of his blessing) on the bread and wine, so that by his power they may become the body and blood of Jesus Christ and so that those who take part in the Eucharist may be one body and one spirit. ...
In the institution narrative, the power of the words and the action of Christ, and the power of the Holy Spirit, make sacramentally present under the species of bread and wine Christ’s body and blood, his sacrifice offered on the cross once for all.
1374 The mode of Christ’s presence under the Eucharistic species is unique. It raises the Eucharist above all the sacraments as "the perfection of the spiritual life and the end to which all the sacraments tend." In the most blessed sacrament of the Eucharist "the body and blood, together with the soul and divinity, of our Lord Jesus Christ and, therefore, the whole Christ is truly, really, and substantially contained." "This presence is called 'real'—by which is not intended to exclude the other types of presence as if they could not be 'real' too, but because it is presence in the fullest sense: that is to say, it is a substantial presence by which Christ, God and man, makes himself wholly and entirely present."
1375 It is by the conversion of the bread and wine into Christ’s body and blood that Christ becomes present in this sacrament. The Church Fathers strongly affirmed the faith of the Church in the efficacy of the Word of Christ and of the action of the Holy Spirit to bring about this conversion.
(emphases added, except epiclesis emphasized in original)
Now, what does Aquinas say about the technicalities of the situation? In other words, if the wine becomes the Blood of Christ, why isn't there hemoglobin in it?
Aristotelian philosophy made a distinction between what (in modern philosophical jargon) is called "substance" (from Latin substantia, the essence or nature of something) and "accident" (from Latin accidens, something that happens to be true of an entity). Aristotle of course used Greek, but Aquinas used the Latin. The substance of something is what makes it what it is: I am a human being because I am human "in substance"; that is, because I have "humanness". I look the way I do as a human because I have particular accidents—my eyes are a given color, my hair and skin, I'm a given height. Any of those things could change, or could have been different; that would change what I looked like, but not what I am (i.e. human).
The Catholic teaching on the Eucharist is that during the Eucharistic celebration, the substance of the bread and wine—what they truly are—is changed into the substance of the body and blood of Christ. But their accidents—what they look and act like physically—remain the same.
In the Summa Theologica, Aquinas devoted several sections to this. The first one (Third Part, Question 75, Article 1) is titled "Whether the body of Christ be in this sacrament in very truth, or merely as in a figure or sign?" After looking at several objections, he concludes:
I answer that, The presence of Christ's true body and blood in this sacrament cannot be detected by sense, nor understanding, but by faith alone, which rests upon Divine authority. Hence, on Lk. 22:19: "This is My body which shall be delivered up for you," Cyril [i.e. Saint Cyril of Jerusalem] says: "Doubt not whether this be true; but take rather the Saviour's words with faith; for since He is the Truth, He lieth not."
Now this is suitable, first for the perfection of the New Law. For, the sacrifices of the Old Law contained only in figure that true sacrifice of Christ's Passion, according to Heb. 10:1: "For the law having a shadow of the good things to come, not the very image of the things." And therefore it was necessary that the sacrifice of the New Law instituted by Christ should have something more, namely, that it should contain Christ Himself crucified, not merely in signification or figure, but also in very truth. And therefore this sacrament which contains Christ Himself, as Dionysius says (Eccl. Hier. iii), is perfective of all the other sacraments, in which Christ's virtue is participated.
Secondly, this belongs to Christ's love, out of which for our salvation He assumed a true body of our nature. And because it is the special feature of friendship to live together with friends, as the Philosopher says (Ethic. ix), He promises us His bodily presence as a reward, saying (Mat. 24:28): "Where the body is, there shall the eagles be gathered together." Yet meanwhile in our pilgrimage He does not deprive us of His bodily presence; but unites us with Himself in this sacrament through the truth of His body and blood. Hence (Jn. 6:57) he says: "He that eateth My flesh, and drinketh My blood, abideth in Me, and I in him." Hence this sacrament is the sign of supreme charity, and the uplifter of our hope, from such familiar union of Christ with us.
Thirdly, it belongs to the perfection of faith, which concerns His humanity just as it does His Godhead, according to Jn. 14:1: "You believe in God, believe also in Me." And since faith is of things unseen, as Christ shows us His Godhead invisibly, so also in this sacrament He shows us His flesh in an invisible manner.
In other words, he says:
- Given that we have faith in Christ, it's incumbent on us to have faith that when he says "This is my body", we are to believe that he is giving us the truth, and not just a signification.
- It is more characteristic of Christ's love for us that he would find a way to actually be with us, not just to be represented among us.
So, yes; Catholics believe that the bread and wine are substantially (in a couple of different senses) transformed into the Body and Blood of Jesus Christ. This belief is part of a longstanding interpretation of Scripture, and writings of the early Church fathers. Catholics do not, however, consider themselves to be cannibals, because the "accidents" of the bread and wine (the ingredients, the flavor, the shape, and so on) are not those of Christ Himself.
I could go into the details of what potential objections to the belief Aquinas considers, and how he responds to them; but that would be matter for a different question.