To understand his words, though, we have to first look at the Hebrew understanding of remembrance. For them, it is not a matter of thinking back to a past event. Remembrance is actually more accurately entitled re-presentation for it is making something in the past present to you once again, making it live again as it were. We see this especially in the celebration of the Passover festival, when everyone who celebrates it is reminded that it is not just their ancestors which have been freed from Egypt, but themselves as well.
In celebrating the Eucharist, Catholics remember that it is not merely those people who lived during Christ's time on Earth who were set free by His sacrificial offering, but that we too have been freed by it. We do not re-create the day Jesus died on the cross for our sins, we make it present so that it is happening right then at the same time as it is happening in the past. It is not possible for this to be done by human beings, but it is entirely possible for a God who is not bound by time to be present in both time periods at the same moment. It is one of the mysteries of the Christian faith.
Christ instituted the Eucharist, giving us his living flesh to eat and his living blood to drink because only that which had in it eternal life could ever possibly nourish us to eternal life. He died to prove to us that death had no hold over Him, and that He was in control of all things. He took a symbol of shame - the cross of slavery - and turned it into a symbol of self-sacrifice and freedom. When we receive the Holy Eucharist, we receive His power, the power to overcome death, the power to transform shame into self-sacrificial freedom, the power to transform our lives, by uniting ourselves to Him. We literally become what we consume.