Let's say we own a company with 10 shares, I own one share, you own 4, and 5 are owned by others. Each share is worth $100 (to make it simple).

The company has $100 "to spare" - they could pay a dividend (give me $10, you $40, $50 for "the others") - but they'd be taxed on the income they made to be able to pay this, and you and I would be taxed receiving the dividend. We'd net out maybe $8, maybe $7 per share.

Or they could buy my share for $100, and retire it. I get the $100 (and pay capital gains tax unless it was in an IRA or otherwise not an issue). You now own 4 shares of a 9 share company, which is worth the same, but your percentage is a big bigger now.

Getting rid of the double taxation of dividends would likely slow down or end most buybacks; the main advantage is that they let the shareholders decide if/when they take the tax hit.