I've blogged about this frequently, and find the two Papal Masses conducted in Washington National Park Stadium and St. Patrick's Cathedral in New York another opportunity to explain one of the most important insights to understanding the Catholic faith and liturgy.
During these two Masses people had an opportunity to see the section of the Mass called the presentation of the gifts. This is when people bring to the Holy Father items that will be used in the Holy Eucharist part of the Mass. So even though it is called "gifts" it is not the bringing of presents. It is called gifts because they are gifts for God.
This part of the Mass is the Christian interpretation of the uninterrupted worship of God through the performance of sacrifice on his altar, just as taught by God to the Israelites and documented in the Jewish Torah and the Christian Old Testament.
God, starting with Adam and Eve, and brought to detail with Abraham and Moses, taught humans how to perform sacrifice to him. The Old Testament details the rules for sacrifice and the forms it may take (animal, grain, oil or monetary sacrifice) and on what occasions it is required. In the Gospel we read how Joseph and Mary brought the baby Jesus to the Temple in order to sacrifice two doves as required. The first murder, that of Abel by Cain, occurred because Cain offered incorrect sacrifice to God and was jealous of the perfection of Abel's sacrifice. So the Israelites conducted sacrifice to God with precision and piety for thousands of years. They stopped doing so when the Temple was destroyed in Jerusalem, in the years after Jesus had died and risen again. So the Jewish "liturgy" does not contain the sacrificial rites that they had for thousands of years.
Muslims have retained sacrifice, once a year during the Haj. They have the direction and permission by God to designate their offerings and the delivery of these sacrificed animals as food for the poor.
Christians retained the instructions to sacrifice on the altar to God! This is something many do not realize, since modern religious instruction and formation has been so inadequate and ill informed. Christians never "dropped doing the sacrifice" because they followed the instructions of Jesus Christ about how to continue performing the sacrifice, but under the New Covenant.
Poor Israelites sacrificed a bowl of grain, since they could not afford a sheep or a bullock. It is mindful of this that Jesus Christ taught that he is now the bloodless sacrifice, in the form of wheat grain baked in bread, in the sacrament of the Holy Eucharist.
Humans cannot afford the price of their own salvation; it is far too high and out of reach. Only Jesus Christ could be the perfect sacrifice for the "poor," who is all of humanity. He therefore instructed very clearly in the Gospels that the sacrifice of bread to God is to continue to be done, with him being the Real Presence in the bread.
He lost some disciples when he first explained this, since they thought of it as cannibalism! Ironically, these Jewish disciples would continue to splash the blood of slain animals (bought through the money changers in the Temple), yet could not accept the sacrifice of the Real Presence in the bread and wine of the Holy Eucharist. This is part of how Jesus showed that the elite of the Jewish faith had lost connection with the poor. Disciples who could afford blood sacrifice were shocked at the idea that Jesus would put himself into poor grains of bread as offering to God on behalf of humankind for their salvation.
So the Christians who stayed with Jesus understood what he said and did at the Last Supper, especially after his resurrection. Therefore you need to understand that the Catholic Church is actually the only faith of Abraham that continues to conduct sacrifice to God, as instructed by God, and reformed in the New Covenant by Jesus Christ.
You can now appreciate the incredible solemnity and continuity, therefore, of the procession of gift bearers in the Papal Masses, who are bringing the instruments of the sacrifice to the altar for the Holy Father to use in the sacrifice of the Mass. They are following the footsteps of Joseph and Mary when they brought Jesus to the Temple with two doves to sacrifice to God in gratitude for the birth of their son.
I hope this helps. I am astonished that people do not understand this great miracle and marvel of the Catholic Church.