What is the special significance of prayers said in a hushed voice by the celebrant during the Holy Mass celebrations of Catholic Church?
First of all, I would like to clarify one point. This tradition is not used across all Rites. In fact St. John Chrysostom makes note that in his Rite, the priest says the consecration in a loud voice.
What started out as a necessity in Rome, turned into an ecclesiastical custom and eventually became the traditional way of saying the Mass of Pope St. Pius V in the West. In this sense we can understand why some priests carry on with this tradition in the Ordinary Form of the Mass.
The very Early Church in Rome originally said its’ Masses in Greek. Latin did not come into the liturgy until around the third or fourth centuries.
When did we start celebrating Mass in Latin?
Mass in the Catacombs was said in Greek. When the Roman vernacular was introduced, the Liturgical Language of the Mass was changed to Latin but things like the Kyrie and some Good Friday antiphons were late introduced into the Roman Liturgy.
Some writers say the Canon is thus said in a low voice because of the persecutions of the Roman empire, and that then they said Mass in secret places and in a low voice lest enemies might hear them. - How Christ Said the First Mass, p. 387 (Rev. James Luke Meagher)
The Mass in Rome being thus said in a low voice originally was brought about by necessity. What started out a means of not getting caught by the Roman authorities simply turned into how the Church in Rome and in the West developed into a Roman custom after the persecutions were ended. This liturgical custom then was carried over into the tradition of the Latin Rite when Latin was introduced into the Roman Rite, while the the use of Gregorian Chant slowly made it’s headway into the liturgy of the West.
The earliest Christian altars were of wood, and identical in form with the ordinary house tables. The tables represented in the Eucharistic frescoes of the catacombs enable us to obtain an idea of their appearance.
The catacombs were underground rooms and passageways that served as mausoleums in which the ancient Romans buried their dead. When the persecutions began and intensified under Nero, the Christians found that they could retreat into these labyrinthine networks of tunnels to escape the notice or pursuit of Roman soldiers or citizens wishing to turn them in to the authorities. The Early Church developed a vast support network and series of hiding places based on the catacombs. Meanwhile, the use of a public mausoleum as a hiding place caused wild rumors about Christian rituals and practice to spread amongst the Romans... Another rumor was that the Christians drank human blood. This may have come from a misinterpretation of Jesus' command: "Behold, This is my blood. Take ye and drink in remembrance of Me." - The Catacombs: A Place of hiding and Worship For the Early Church
During the 3rd century, Christians actually used to worship within the catacombs. Traditionally they would worship at home, but as the religion grew, they needed more space. Benches and tables were taken down to the catacombs for faithful worshipers to pray among the dead. On a Rome catacombs tour, you can walk among the spaces that very early Christians used for worship, prayer and community.
The catacombs run deep - they are at least 20 meters underground, and many are around 20 kilometers long! The catacombs are actually just outside of the city center: it was illegal to bury the dead within the old city walls, likely for hygiene reasons, so you do have to travel to visit the Roman catacombs.
There are well over forty catacombs under the city of Rome. However, only five of these are open to the public. - 10 Haunting Truths about Rome’s Crypts and Catacombs
Even St. John Chrysostom admits that the tradition of saying the Canon of the Mass aloud is an acceptable tradition as is found in the Eastern Churches. In fact, the Maronite Catholic Rite, which has never separated from Rome has always prayed the consecration aloud and with the congregation in singing dialogue, except the words of consecration itself which are chanted by the priests in Aramaic, the language spoken by Our Lord to the Apostles. How beautiful! I imagine that in times of persecution, even the Eastern Churches would say Mass in a low voice in order to avoid getting caught by the enemies of our holy religion.
We do not deny that the words of consecration in the Eastern Church are customarily recited out loud, since it is quite certain from the liturgy of Chrysostom. Nor do we condemn this; for we do not contend that these words must necessarily be recited in a quiet voice, rather, that the Church is free to establish the rite, and hence neither the rite of the Greeks nor of the Latins can be condemned, nor should they. But although the words of consecration are uttered in a loud voice among the Greeks, nevertheless, certain others are pronounced in a quiet voice and clearly in secret, as we clearly showed from the same liturgy of Chrysostom. - On the Most Holy Sacrifice of the Mass, ch. 12 "Not everything in Mass must necessarily be said in a loud voice" (St. Robert Bellarmine)