In the history of Christian thought, the question of why Adam and Eve were naked has been more significant than one might think. There have been a couple of options. First, Augustine didn't really have a positive explanation for it. Negatively though, he argued in The City of God book 11, that it was possible for them to be naked because of the lack of lust within them. Prior to the Fall, he argued, there would have been a perfect self-mastery and composure. This means that people wouldn't have really been all that interested in sex in and of itself. It would have been a something purely interesting for the sake of procreation. Being the Platonist that he was, he generally thought that erotic desire was best directed towards "the Good" which he of course identified with God. After the Fall, our first parents had to put on clothing to hide the visible manifestations of the lust the felt for one another. The human person became a microcosm of the macrocosm. Just as the universe was in revolt against God, so too our bodies are now in revolt against us.
The second option is that of some recent biblical interpreters who seem to follow an Irenean understanding of the Fall, though from my reading of Ireaneus, this is not a position that he explicitly took. In this understanding, prior to the Fall human beings are like children and therefore, much like children in a lot of the ancient world, do not wear clothing. In this scheme, the Fall leads to us putting on clothing and becoming adults. Hence sin for some of this commentators leads to greater moral maturity and agency.
Both of these position are of course problematic. Augustine is of course correct about the lust of the flesh and the fact that our bodies are fallen, but he is wrong (as Melanchthon notes in the Apology) to think that the desire of man and woman for one another is from the Devil. Similarly, the view that Adam and Eve are like children has all the same problems that both Irenaeus and Hegel have. By making Adam and Eve childlike, Irenaeus did come up with a rationale for the Fall, but he also made us less culpable. Hegel made sin necessary for our self-consciousness and for God's. Hence, it again is not that blame-worthy. It is a felix culpa on steroids.
In order to answer this question in a more satisfactory manner, let us examine the concept of "bodies." When he's not developing strange tritheistic Hegelian heilgegeschicte schemes, Robert Jenson has developed an interesting definition. Many different things can be bodies, of course. Within the human world, in which we are relational beings, who have both a spiritual and physical natures. Since we are inherently relational, our bodies are our personal availability to one another.
In light of this definition (which to me seems to cohere well with the biblical understanding of human ontology), the nudity of Adam and Eve take on a meaning different from earlier ones we encountered. If Adam and Eve have received all things from God as one who gives them life and freedom (Gen. 1:28), then they are nude because of their perfect freedom to be available to one another. As "lords of all," they are capable of being servants to one another. In other words, within the freedom of perfect faith, they are totally free to completely surrender to another and therefore be perfectly available.
Sin is what makes them clothe themselves. The Bible describes them as being ashamed. This coheres with their need for self-justification under sin. They cannot believe themselves to be "very good" any longer. They must conserve themselves and hold themselves back due to their sin. Without the perfect freedom of faith, they cannot be perfectly available to one another or to God. They must be in rivalry with one another and they must self-justify before God. They must in a sense "cover" themselves before his judgment.
Christ reverse this in the nudity of the cross. He first fully exposes the depth of human sin, both by dying at the hands of sinful men. Their need to reject God's judgment and thereby self-justify is so strong that they must go so far as to kill God himself. On the cross, Jesus also reveals God reaction to sin. There is no more covering up before divine judgment as in Genesis 3. There is full exposure before all.
At the same time, he makes himself as a redeemer from sin perfectly available to all by exposing his body to all: "when I am lifted up, I will draw all men to myself." He continues this in the sacrament of his body and blood, "this is my body given to you for the remission of sin, etc." He thereby creates a new tree of life on the cross and reverses the Fall.