The central and still startling claim of the Catholic Church is that Jesus is really, truly, and substantially present under the forms of bread and wine. His presence is not simply evocative and symbolic, not simply the result of our thinking so or wishing so, but rather real, true, and substantial.
If you want to find this verified scripturally, look of course at the accounts of the Last Supper in Matthew, Mark, and Luke - and also in Paul. But look especially at the sixth chapter of John's Gospel. Jesus identifies himself as the "living bread come down from heaven," and then he specifies, "Unless you eat the flesh of the Son of Man and drink his blood, you have no life in you."
Now this was extremely objectionable language for a Jew of Jesus' time. To eat someone's flesh was a term of contempt. More to it, the drinking of an animal's blood was expressly forbidden throughout the Old Testament - much less the drinking of a man's blood. But when Jesus' listeners object, Jesus does not soften his language - he intensifies it: "My flesh is real food and my blood real drink."
How can we make sense of this claim? It has everything to do with who Jesus is. If he were simply an ordinary human being, his words would have, at best, a symbolic resonance. I can say, "This ring is a symbol of my love for you." But Jesus is God, and what God says, is.
God's word affects reality at the most fundamental level. Thus, when Jesus' words over the bread and wine are spoken, they change into what the words signify. They become really, truly, and substantially the body and blood of the Lord.
The reality of this sacrament so important because "For whoever eats my bread and drinks my blood has eternal life." The Eucharist, as the eternal presence of God, eternalizes those who consume it, making us ready for eternity. We participate in Jesus Christ through this sacrament.