It's a way of "giving money back to shareholders" because buying their own shares on the open market means that they inflate the stock price on the market by buying the existing sell orders out there.
A buyback doesn't inherently increase the price of the shares. A buyback works like this: Start with a company with an operating business valued at $750M and a pile of cash in the amount of $250M. You own 100 shares out of 1000 (10%). Your shares are worth $1B * (100/1000) == $100M. The company now does a buyback of $50M. They buy 50 shares from other shareholders and cancel them. You now own 100 shares out of 950 of a company with an operating business valued at $750M and a pile of cash in the amount of $200M. Your shares are worth $950M * (100/950) == $100M.
That doesn't mean the share price won't change in practice, but not because of the math, rather because of what investors make of the decision.
