Women are displayed with exposed breasts in Minoan artwork from 1500 B.C. Some historians believe that these ancient women went topless only during religious rituals—bare-breasted, buxom goddesses have been worshiped since the dawn of civilization—but some of the artworks depict everyday activities, suggesting that bare breasts may have been commonplace. Just across the Mediterranean, ancient Egyptian women sported elaborate dresses that could either cover the breasts or leave them exposed, depending on the whim of the designer. Over the next few centuries, however, breasts become strictly private parts. Ancient Athenian women were wearing flowing, multilayered robes that concealed the shape of the bosom by the middle of the first millennium B.C. Spartan attire was more risqué, exposing the female thigh, but breasts were always covered.
Covering the breasts suggests a greater sense of self-awareness and perhaps self-consciousness. In many primitive cultures, the breasts are seen as just another utilitarian body part, and women are not self-conscious about them, and men do not make remarks about them, even in a male-only setting. The only universal taboos about body exposure have to do with the genitals and the anus, which unlike the breasts are involved in excretion and intercourse, two of the most obviously shameful and/or private activities.
The fact that Mediterranean peoples before 1000 B.C. resembled primitive cultures, and that they had radically changed toward greater self-awareness by circa 500 B.C., makes me immediately think of Julian Jaynes' ideas about the bicameral mind. He was trying to account for an apparent lack, or at least a very low degree, of self-awareness in civilized people before around 1000 B.C., and their shift toward introspective abilities and self-awareness by around 500 B.C.
His main sources for documenting this are comparing the older Iliad to the younger Odyssey, and the older books of the Old Testament like Amos to the younger ones like Ecclesiastes. In short, characters in the older literature seem to be trapped in concrete awareness of the world around them, with little or no abstraction or introspection, whereas their counterparts in the younger literature can introspect and ponder questions about human nature, divine nature, love, hate, and so on.
Older religious traditions do not assume that a group member could look inside themselves -- they received a message from the gods about what to do, and they obeyed. However, looking inside yourself is the central feature of all of the major world religions and philosophies that were born and began spreading independently all over the Old World during the Axial Age, such as Second Temple-ism, Zoroastrianism, Buddhism, Platonism, and Confucianism.
He also discusses how the Greek language of the Iliad had no abstract mentalistic terms like thought, soul, anger, etc. They only later acquired such abstract meanings, and originally had very concrete sensory meanings like sight, heavy breathing, excitation of the nerves, life-matter, etc. Ethnographers of hunter-gatherers also say that the people they study rarely or never at all hold abstract discussions, even about the everyday world. For example, they don't say "it's a beautiful day," referring to something abstract like "beautiful." They phrase it more concretely, like "it's hot out" or "the sun is shining."
Jaynes' approach is more interesting than comparing different cultures today because he's tracking a change over time within a population. That way we can better see which things are associated with which other things -- if they're tightly related, they'll show similar trends over time. If we just compare different cultures today, it's more difficult to disentangle which differences are related to which others. There are "spurious" correlations that don't reflect a true relationship deep down.
For example, the Bushmen hunter-gatherers speak a language with clicks, whereas no modern language has clicks -- but that could be unrelated to other psychological differences between them and us. Now, if we found out that the Minoans spoke a language with clicks, and that they were lost over time by circa 500 B.C., that would squarely place linguistic clicks within the suite of traits leading from primitive to modern minds.
So, at least to judge from the available record of ancient visual culture, it looks like modesty about the breasts was another piece of the larger shift toward self-awareness that took place during the first millennium B.C. across large swaths of the Mediterranean and Middle Eastern civilizations. Before, only the most concretely shameful body parts were the targets of taboo-related behavior -- covering the genitals and anus.
As the ancient Greeks came to develop greater introspective and abstract thinking, they probably debated with one another in everyday settings whether or not other body parts ought to be covered. The breasts are not involved in shameful activities per se, just nursing infants, and then only occasionally. But they are a physical difference between the sexes, they begin to develop during puberty, and they're an orifice through which liquid leaves the body.
To all primitive peoples, this strained line of reasoning doesn't occur to them, and even if they're told about the civilized pattern of covering them up, they reject it as a ridiculous and perhaps painful practice. But to people who are becoming more self-aware and abstract in their thinking, the breasts emerge as the next obvious target for body exposure taboos.