Here is Cicero from De Legibus:
“For it appears to me that among the many exceptional and divine things your Athens has produced and contributed to human life, nothing is better than those Mysteries. For by means of them we have been transformed from a rough and savage way of life to the state of humanity, and been civilized. Just as they are called initiations, so in actual fact we have learned from them the fundamentals of life, and have grasped the basis not only for living with joy, but also for dying with a better hope.”
So Greece created democracy, philosophy, and whole lot of great art, but the thing the Romans appreciated most was whatever sort of experience they had at the temple of Eleusis outside Athens?
Lots of ancient sources talk about whatever it was that they were doing there vaguely. An inscription from Eleusis reads, "Beautiful indeed is the Mystery given us by the blessed gods: death is for mortals no longer an evil, but a blessing," so the experience seemed to take away a fear of death.
When the Christians started trying to destroy the temple and end the Mysteries, Vettius Agorius Praetextatus, who had been initiated at Eleusis, said destroying the rites “would make the life of the Greeks unlivable” and argued that the temple was the one place that “holds the whole human race together.”
Marcus Aurelius was no spendthrift, and not given to flights of fancy, but when the temple of Eleusis was destroyed by a barbarian raid into Greece, he paid to have it rebuilt immediately, and went there himself.
Do we know why Cicero and so many others ranked the experience they had at Eleusis so highly? Why did they think it was so important to keep it going? Why did they view it as a civilizing force? What exactly went on there?
The mysteries of antiquity are a bit of a specialty and I spent my undergrad thesis devoted to them especially Eleusis and Dionysus. Allow me if I may to attempt to present to you an approach of an answer although the interpretations below are mine based on the primary and modern scholarly evidence. Further reading at the end of post. First we will never know what happened behind the walls of the Telesterion. The initiation rite was a secret oral tradition that was guarded fiercely with only scanty Christian apologists giving portions of the ceremony in the third century C.E. The heart of your question was why Eleusis held such a deeply prestigious place in Mediterranean antiquity. The answers vary but three are of importance. The first is its international character by design centralized on its geographical location. Eleusian mysteries were held in the telesterion in Eleusis and people came from a far to take place in them. Although at most times Citizenship and inclusion in society was reserved for very few citizens the mysteries, according to Herodotus and other primary sources,were open to all with the exception of those guilty of 'blood crimes' i.e murder, and those who did not speak Greek. The Eleusian mysteries were unique for religious inclusion. Xenoi, and slaves with advocates of Greek stature were even allowed so long as they paid the fees( about one full day of work for the time). But why travel to be initiated? That brings us to the second reason. The mysteries at Eleusis were unique along with Dionysian/Orphic for eschatological and soteriological concerns. One must remember this was pre Christian and an afterlife that promised more than meandering or darkness was a welcome comfort for average citizens. Finally the mysteries would have a Cathartic tension release valve revolving around societal and economic standing especially closer toward the 4th century. On the bridge to the sacred well the day before the mysteries proper, from Aristophanes at least, we see slaves hurl insults at the wealthy, the elderly lampooned by the young, flashes of nudity, and wealthy female against female infighting. Why? This was a spiritual cathartic release. A type of blood letting before the ceremony. One washed themselves in the sea afterward as well. Clean body clean spirit. When we look at the mysteries and religion in general in Greece under the guise of what Servinou-Inwood calls polis religion,rightfully so, and the imperial/republic cult of Rome we see a culture different from our own. Secularization was not natural in the classical world. Religion, daily life and politics were integrated seamlessly within society from beggar to emperors.
In the end the mysteries at Eleusis were important as a religious cultural giving structure to the Greeks and later to the Romans who borrowed it from them. I could type out an entire essay on this but its late. The short answer is the eschatological, social cathartic, and cultural structural stability the Eleusian cult gave the Mediterranean world until its fall that made it so important and worth protecting.
If you have anymore specific questions feel free to ask.
*An aside: I know the tendency to describe Eleusian mysteries as psychedelic is a appealing however there simply is no good historical evidence available that they utilized psychotropic drugs to get their desired effect. Ergot is usually cited through the initiation drink Kykeon however the amount needed for hundreds of initiates as well as conscious utilization of cannot be verified.
Further reading: Christiane Servinou-Inwood, "What is polis religion", and " what is polis religion now" Jan Bremmer, " Initiation into the mysteries of the ancient world" Hugh Bowden, " mystery cults of the ancient world" Herodotus "Histories"