How did the modern hyper-appreciation of antiquities develop?

This is a good question. I don't have any hard sources, unfortunately; but I can speculate a bit.

If we speak of the extremes to which care and protection of cultural artifacts have been taken today, I would think that the most important reason is reactions against the somewhat careless archaeology of the 19th and early 20th centuries. Seeing how much was lost in the hasty and sometimes amateurish work probably prompted measures to curb careless excavations, which may well have led to further compounding of a general sentiment for a "prerogative", so to speak, of properly scientific archaeologists. I admit this explanation alone is somewhat incomplete without an answer for why archaeology "boomed" so fast in the 19th century, though. I can only guess this was because the rising living standards of the middle class allowed more people to travel abroad and visit ancient ruins. Another contributing reason could be that the growing colonial empires of the West allowed western subjects to travel further and longer while still under greater security and a reasonable protection of law.

If we speak of the modern prices more specifically, this is easier to answer. Since many (all?) nations have adopted laws for the strict preservation of archaeological artifacts, the supply of such artifacts available for purchase has been very greatly diminished, obviously driving up prices a lot.

It can probably also be said that the active practice of archaeology as a discipline has only started to be possible due to the general increase in wealth since the middle ages onwards. Archaeology is a rather specific science which is hard to practice without the specialization allowed for by a more advanced division of labor, which can only come from greater capital accumulation.

This is all speculation, of course. If I were to try and find sources, I would look for preparatory investigations and/or parliamentary debate for the first laws which regulate trade in antiquities. I don't know where to find such, but it would probably reveal why such laws were enacted, and shed some light over the sentiment of the day.

In addition to the other answers, here's a psychological bit:

People today, especially Americans, live in a faster age, with much shorter time scale of things. We build houses in 1 year and don't count on them to survive beyond 20 years. We use gadgets designed to survive 2-3 years before being replaced. This greatly amplifies our appreciation for the things that survived for centuries, sometimes millennia.

A person who lives in a 100-year-old house finds it natural that things survive for a very long time. When I first came to California from Russia I was very surprised that some buildings that were younger than my grandfather were classified as "heritage" and new owners were precluded from modifying their façades. "Who cares that this average looking building is 70 year old? The publisher across the street back in Ukraine occupied a 150 year old building, and some neighbourhoods are much older than that" - I thought.

My classmates studied archaeology back in Russia, and he told the stories of excavations where literally tons of artefacts were being thrown into trash due to lack of originality. Classification on that site in Veliky Novgorod went on like this: if the 12th century vase was perfectly preserved then it goes to the museum; if it's chipped then it will be sold to the tourists; if it's broken then it goes to the trash. Mind that these vases were 3 times older than the United States, and the excavation was performed in mid-1980s. More care was taken of the birch bark scribbles though.

Old World just used to live on a time scale very different from American one. The abundance of history all around you made made antiquity psychologically less remote.

The timing of the renewed appreciation of antiquity that you suggested coincides with the rise of the USA as the dominant world power. That rise to power projected some of the American mode of thinking, apparently including Americans' time scale. That would explain the change of the attitudes toward antiques in the parts of the World that were dominated by USA, and lack thereof in the part of the World dominated by USSR, where time remained slow paced enough to deem 800 year old artefacts routine and boring.

There's really a few questions being asked and it would be better to separate them.

Why didn't Europeans in the Middle Ages appreciate "antiquities?"

First, because archeology hadn't been invented yet and two because it was the Dark Ages. Now, the Coliseum was scavenged because so much knowledge was lost after the collapse of the Roman Empire, the people really needed the material, but there other things at play as well. During the early Medieval Period, or "Dark Ages," through the Renaissance, art was regulated by the Catholic Church. A good amount of very beautiful art was created, but its intentions were to help its audience spiritually. All writing, music, painting, drama, sculpture and architecture created during the period reflect these notions. What was art and was not art was something decided by religious clergy. They could be particularly hard on dramatic performances, multitonal music or any developments in art. At the time, they absolutely did not consider pagan philosophy or art to be appropriate topics.

In the Renaissance (14-17th century), the attitude lightened a great deal. Pagan philosophy and ideas were taught in universities. Nonreligious topics were allowed in art again for the first time. The movement originated in Italy and there was also interest in studying the old remains of classical Roman buildings.

Filippo Brunelleschi was foremost in studying the remains of ancient classical buildings, and with rediscovered knowledge from the 1st-century writer Vitruvius and the flourishing discipline of mathematics, formulated the Renaissance style which emulated and improved on classical forms. Brunelleschi's major feat of engineering was the building of the dome of Florence Cathedral.[52]

Why could antiquities be purchases so cheaply, so recently?

Short Answer: Because they were stolen (or collected) from nations that were European colonial possessions. Also there was a misunderstanding of the best ways to perform archeology in general.

Long Answer: Europe emerged rather quickly from the Renaissance with several nations competing for global empires. These developments influenced European art and philosophy. They were very curious about other cultures and thus the European aristocracy and nobility tried to add the best artwork from around the world to their private collections. In some instances, such as Napoleans conquest of North Africa and Egypt, the art and antiquities were outright stolen through war. In other instances, they were likely gifts, although as you will see below, today it is not considered possible to actually gift a nation's most prized artwork.

So, the influx of such large amounts of foreign antiquities made the price lower than today. Also, now that these colonies are independent nations, they have rightly requested back their artwork, so it is no longer on the market. This increases prices a great deal.

As to the second point, I believe Dolda2000 answer is satisfactory. It simply isn't allowable to sell artifacts piecemeal as was once done, since it ruins the context for the artifact. Research is much more profitable if the researcher has all the artifacts and if they haven't been tampered with. Therefore, the market for antiquities is again restricted.

Why are they consider "priceless" now?

The modern appreciation and value placed on historical and artistic artifacts is derived from the creation of the modern nation state. In order for a people to understand who they are, they must have a preserved shared past. Examples from the late Middle Ages and early Modern Era in France and Germany will be instructive. In modern times, many (if not all) artifacts are now literally considered priceless (even if sometimes they can be purchased at a price). This is because they belong to "the people" and "posterity" and this idea comes from the French Revolution. The price for private collectors is driven up since most artifacts will be in public museums.

The French Revolutionaries converted Louis XVI palace into the famous Louve museum. While museums did exist before this, they were somewhat rare. This action was a statement that the rich and powerful do not have the right to keep art for their personal enjoyment. It belongs to us all. This idea drives laws and restrictions on purchases on art and antiquities.

Source: Sherman, Daniel J. and Rogoff, Irit, ed. Museum Culture Histories Discourses Spectacles