Although it may not be the easiest thing for modern ears to interpret, the signifance of Chaucer's work cannot be overstated. For one thing, Chaucer championed the vernacular in his work, that is, the real language of everyday citizens. Prior to Chaucer, works of literature were written in Latin or French; while there certainly were English writings, they did not make use of the vernacular. Chaucer's tales helped bring meaningful stories of life to the everyday man.
Furthermore, due to Chaucer's efforts, scholars are able to discover much about the past through his stories. The people Chaucer depicts, their ways of life (work, love, sex) were not the stories recorded by the elite. We are, therefore, able to gain insight into the middle and lower classes which might have otherwise been lost.
There are many reasons that this is an important piece, but one that truly stands out is that it was the first book of poetry that was written in the English language. This made it available to many readers--not just the elite. It opened the doors for others to be written in English, as well. It is a record of the roots of our language and how it has evolved.
This piece of literature is the very first of its kind. It is the first collection/anthology of short stories as we know them today. Chaucer brilliantly gives us an amazingly accurate cross section of society in the Middle Ages, and if you look hard enough (and sometimes not so hard) it is easy to find modern day equivalents to his characters. He also gives us a psychological view of humanity, exploring our motives for lying, cheating, stealing, and so on. Who among us can say we are perfect? The only truly good characters he introduces are the knight, the parson, and the plowman. Every one of them is corrupt in some way-- not unlike the modern ages-- and they all justify their actions. It's a brilliant look at people of all ages, occupations, beliefs, values, and social status. Not only does Chaucer tell us what he notes about them, but they also reveal much about themselves through the stories they choose to tell. So cool!
One reason Chaucer's work has stood the test of time and is still relevant today is that it illustrates an interesting cross section of people from the Middle Ages. He authentically captures the human spirit inherent in people from all walks of life from that time period. Just look at how complex of a character the Wife of Bath is. She would fit perfectly into our society today. And for those who think our culture is too vulgar and degenerate, just read "The Miller's Tale." These are but a few reasons why The Canterbury Tales serves as an important piece of literature.