Italo Calvino, the great Italian author, had a go at answering this question a couple of decades ago, and the resulting essay is a great read if you haven't seen it already. Before reading his essay, I wouldn't say that I was antagonistic towards the classics, but I was far from being agonistic by any means. If the classic is fictional, then it will inevitably be written in impenetrable language and be full of stuffily ancient viewpoints. If it is non-fiction, then surely the subject matter will have far advanced since the time of the author. But then, I dipped my toe in and I discovered something quite odd - classics were often surprisingly good.
Before we go any further, we should probably work out what a classic is, exactly. Calvino proposed several definitions for 'classics' - but one has stuck with me for some time. He states, quite clearly, that a classic is a book which integrates itself into your consciousness, and influences your decisions (for the better, I would hope) somewhat unconsciously. And after trying out a few classics, I can quite happily verify the statement. Whitehead's 'The Aims of Education' sounded like a tedious philosophical diatribe, but it turned out that it answered one of the most common questions about modern teaching - what is the point in learning things if they will never be useful? From then on, I have asked myself why on Earth a quadratic equation could have a purpose, what a Hertzsprung-Russell diagram does for society, how triplets in writing are as persuasive as they are. And I can honestly say that my experience of education has been the better for it.
Another reason is that classics are the foundation on which our culture is built. If you understand how culture was built up, what people really thought about the universe, why politicians have always acted the way they do, then you will be in a much better place to understand the world around us today. The reason these books are still read is that they provide the skills and critical understanding that every citizen should have. If you read Euclid's Elements, then I admit that you will probably not be called upon to draw rectilinear angles of a given size. But the way it trains you in logic is simply unbeatable. Equally, Machiavelli's The Prince is partially a short history of Florentine political infighting in the 15th century, but it is so much more than that - it exposes the methods that politicians can use to control an unruly populace. We are still the same people that we were a thousand years ago. Human nature has barely changed. And this is why the classics are still relevant.
My final reason for reading the classics probably applies more to fiction rather than non fiction, but it is important nevertheless. It is that most classics are genuinely good books. When I first started the Odyssey, I expected a drab tale of some people chilling on a boat. What I actually experienced was a genuinely gripping story of one guy's struggle to get home after a war. When I read Cicero's On Old Age, I really liked it, not because it appealed to any high-and-mighty philosophical beliefs, but because it reassured me that old age is something to look forward to, and that death is not something to be feared. Admittedly, it's hard to become even remotely intrigued when Adam Smith discusses what once happened to the price of corn for three chapters, but even there, there are some really interesting points made that I can't help but see in everyday life.
Overall, classics have helped me in a way the education system never has - they have given me a good understanding of the way the world works, as opposed to reams of unsubstantiated facts and algorithms. I urge you to have a go - you certainly won't be disappointed!
A reading list
There's a lot of 'great book' reading lists available online - I specifically recommend Mortimer Adler's 'Great Books of the Western World' and 'Gateway to the Great Books' - so I won't bore you with another. What I will suggest is a collection of books and essays that are a good introduction to classics. I've tried to avoid anything overbearingly long, and I've used a lot of excerpts. If any take your interest, use these selections as a springboard for further reading. (This is, of course, a work in progress).
Book I of the Elements, Euclid
The Eruption of Vesuvius, Pliny the Younger
On the Origin of Species, Charles Darwin
A Chemical History of a Candle, Michael Faraday
New Introductory Lectures on Psychoanalysis, Sigmund Freud
A Mathematician's Apology, Godfrey Hardy
Cosmic View, Kees Boeke
A Brief History of Time, Stephen Hawking
Man and Society
The Battle of Hastings, William of Malmesbury
On World Government (Book I of On Monarchy), Dante
Of Idleness, Michel de Montaigne
The Communist Manifesto, Karl Marx and Friedrich Engels
The Universal Declaration of Human Rights
The Road to Serfdom (abridged), Friedrich Von Hayek
On Old Age, Cicero
Meditations, Marcus Aurelius
The Prince, Niccolò Machiavelli
The Aims of Education, Alfred Whitehead
Why Read the Classics?, Italo Calvino
The Odyssey, Homer
The General Prologue to the Canterbury Tales, Geoffrey Chaucer
Romeo and Juliet, William Shakespeare
What Men Live By, Leo Tolstoy
Animal Farm, George Orwell
The Old Man and the Sea, Ernest Hemingway
Most of these are available free online. I'm thinking of setting up a book club for these - PM Krollo if you're interested.