The shares you buy through a broker-dealer are held in "street name" meaning they are registered with your broker-dealer. You enjoy beneficial ownership, meaning you enjoy the benefits of ownership even though the title to the shares are in your broker's name. You have IOU's, basically.
Because the shares aren't in your name, broker-dealers can lend them out to hedge funds shorting the company you're investing in. The shares can also vanish if the brokerage goes under.
DRSing your shares at the company's transfer agent ensures that the shares are registered under your name (and not the brokerage's), and prevents your shares from being lent out and being used against your investment.