For much of the Church's history many of the priest's prayers at Mass were inaudible to the congregation. This wasn't simply because they didn't have microphones or because the priest prayed ad orientem, facing in the same direction as the congregation. It was the style of the liturgy, and you will still experience the Mass this way if you attend the Extraordinary Form.
Even today, however, in the post-conciliar Mass, during which the priest generally faces the people and many parts of the liturgy are prayed responsively or collectively, there are still some prayers of the priest that those outside the sanctuary may not be able to hear.
Why is that?
According to Fr. Edward McNamara, "In the ordinary form of the Roman Rite this quiet recitation is mostly reserved to the priest's personal prayers." Sometimes these prayers are called "'priestly apologies,'" which are not prayers in which the celebrant excuses himself for being a priest, but in which he recognizes his intrinsic indignity and implores divine aid in order to worthily celebrate the august mysteries."
Priest as Intercessor
One example of prayers said in a "low voice" comes during the Offertory, when the priest offers to God the bread and wine. The rubrics, or directions, of the Roman Missal state, "The Priest, standing at the altar, takes the paten with the bread and holds it slightly raised above the altar with both hands, saying in a low voice: 'Blessed are you, Lord God of all creation, for through your goodness we have received the bread we offer you: fruit of the earth and work of human hands, it will become for us the bread of life.'" He also prays a blessing of the wine.
In this instance the priest is acting as an intercessor for the people, offering the congregation's gifts to God and begging his blessing upon the Holy Sacrifice. At Sunday Mass, we generally don't hear these prayers because we are singing the offertory hymn. At daily Mass, however, the congregation can hear the priest's softly spoken words, and responds to each blessing prayer with the refrain, "Blessed be God for ever."
An example of the priest's personal prayers comes after the Offertory, when he goes to wash his hands. The Missal says:
"After this, the Priest, bowing profoundly, says quietly: 'With humble spirit and contrite heart may we be accepted by you, O Lord, and may our sacrifice in your sight this day be pleasing to you, Lord God.' Then, standing at the side of the altar, he washes his hands, saying quietly: 'Wash me, O Lord, from my iniquity and cleanse me from my sin.'"
It is obvious why these prayers are prayed quietly, since they only concern the priest and his relationship with God.
Theses prayers are just a few examples of times when the priest is instructed to pray in a "low voice" or "quietly," directing his prayers to God on behalf of the people.
A few years ago Cardinal Robert Sarah commented about this aspect of the liturgy during a speech at the Fifth Roman Colloquium on Summorum Pontificum (the Motu Proprio of Pope Benedict XVI acknowledging the importance of both the Ordinary and Extraordinary Forms of the Mass), held at the Pontifical University of St. Thomas (Angelicum). According to the Catholic Herald, Cardinal Sarah said, “The silent praying of the offertory prayers and of the Roman canon might be practices that could enrich the modern rite today. In our world so full of words and more words more silence is what is necessary, even in the liturgy.”
Silence is a key part of the Mass, something that the Roman Missal specifically states. It suggests that "brief periods of silence are also appropriate, accommodated to the assembled congregation; by means of these, under the action of the Holy Spirit, the Word of God may be grasped by the heart and a response through prayer may be prepared. It may be appropriate to observe such periods of silence."
Similar to how Elijah heard God not in the wind, earthquake, or fire, but in the “a whistling of a gentle air,” sometimes we need to pause for a second and listen to the silence instead of the prayers being said by the priest. God may be trying to speak to us, and we simply need to open hearts to hear his words.