What is the significance of transubstantiation? During the Eucharist ceremony where the Priest says some words and then transubstantiation takes place, what is the significance of this event?

I realize this is derived somehow from the last supper verse, but what does it mean?

Transubstantiation means the wine and bread literally become the blood and flesh of Jesus.

What does it mean to drink the blood and eat the flesh of Jesus?
Good question! The Transubstantiation is the spiritual AND physical process where the Eucharist changes into Jesus's real Body and real Blood at the hands of the Priest.

Summary

Jesus states at the last supper that "... this IS My Body..." As Catholics, we believe that He is telling us to have faith enough to believe His words. At the first supper and at every Mass held after, the Holy Eucharist has ALL the properties of Jesus's Body, yet has the outward appearance of Bread. By appearances, I mean every physical human observation that it can undergo. Whether that means touch, taste, smell, sight... it will still appear as bread. But, even with these outward properties, the Eucharist IS Jesus's Body. The same goes for the wine. We must have faith that He tells us what is true.

By taking Jesus's Body and Blood into your own being, you take in renewed strength against temptation. It brings closer the fact that Jesus made a sacrifice on the cross so that YOU may strive away from temptation. YOU must accept His sacrifice (the sacrifice of His Body and Blood) by taking Him unto yourself.

Three reasons why Our Lord Jesus Christ gives us His Body and Blood in the Holy Eucharist are:

    So that his Church on earth may receive Him through the sacrifice of the Mass.
    So that all of His people may remember the cross that he bore for them.
    So that all may remember the love that he has for us, and worship Him accordingly.


Personal Research

Click here for a thoroughly informative article on Transubstantiation by Frank J. Sheed

Click here for the Vatican's own explanation of Transubstantiation

You might also check out the Catholicism TV series which gives many stunning explanations on Catholicism in general (IT'S REALLY GOOD)

More on why we partake in His Body and Blood

I hope this has answered any questions you have had. Feel free to leave a comment below if not!

God Bless!