Not that rare at all actually.
With tons of diamonds being dug-up every year, diamonds are among the least rare gems in the world. While they were rare once-upon a time, with the industrialization of their production they ceased to be rare around the 1880s.
The question should rather be, why are they so valuable?
Again, they're not.
The chairman of the planet's largest Diamond producers, the DeBeer corporation, put it like this: "diamonds are intrinsically worthless, except for the deep psychological need they fill."
When the large diamond mining began in the late 1800s, the diamond corporations began stockpiling the many finds and selling them off, bit by bit in small amounts. This artificially created 'limited amount' gave them the possibility to sell the gems at high prices. In the following 50 years the N.W. Ayer & Son advertising company (the same people that brought you the “When it rains it pours”, "Be all you can be”, and “Reach out and touch someone”- slogans) initiated the world’s most successful advertising campaign ever conducted for a product with zilch value. The trick was selling a concept, not a product.
Using Hollywood (Diamonds Are Forever, Diamonds Are a Girl's Best Friend) as well as print media advertisement, they pushed the image that diamonds are rare and valuable. They particularly pushed the impression that an engagement ring needs to have a diamond, and that a man needed to spend an exorbitant amount of money to so prove his love.
Thus the only reason we believe that diamonds are rare and valuable are due to a successful brainwashing marketing campaign.