In a hedged investment, the act of closing one position while keeping the other open. For example, suppose one owns a stock and also holds a put option to sell that stock at a certain price in case the stock price drops significantly. In this case, legging out would involve selling the put option while continuing to own the stock.
The closing of one side of a hedged position while leaving the other side of the hedge position open. For example, an investor might buy an October call on ExxonMobil and sell short a November call on the same stock. Subsequently buying the November call to cover the short position while continuing to hold the October call results in the investor legging out.