The syndicate typically sells the base deal plus the shoe, leaving it short the extra shares. If the stock trades up, the banks exercise the option and buy those shares from the company at the offer price, closing the short and raising more money for the issuer. If the stock trades down, they instead buy shares in the open market to cover, which is syndicate-stabilization.
So the shoe is a two-sided tool: extra supply in a strong deal, a built-in buyer in a weak one.
Example: a 20M share base deal with a 3M share shoe. The syndicate sells 23M shares at $28. Trading above $28, it exercises and the company receives an extra $84M. Trading at $25, it buys 3M shares in the market instead, spending $75M to cover a $84M short.
Related: underwriter, syndicate-stabilization, ipo, ipo-pop, ipo-allocation