You seem to have understood this for the most part. What this really boils down to is simple economics: supply and demand. Apple says they will spend $100 Billion to buy back shares. But who's shares will they buy, yours, mine? How to decide? How does this help shareholders?
Well... as it turns out, people who know about this news will realize that there is an increased demand for Apple Stock (since Apple effectively entered the market to buy their own shares). These people will hold out to sell at a higher price. More buyers means greater value for the product -the stock in this case. Others will follow slowly causing the price to gradually rise, the stock rose ~4.6% after the news broke.
People who want to sell will take the money and sell (at the higher price), while others will hold on, and will be rewarded by a higher share price of their existing Apple shares. Thus, everyone wins!