Theta-Gamma Relationship and Delta-Hedged P&L

You are running a delta-hedged option position -- you own the option and continuously adjust your stock hedge to stay delta-neutral. Show that the daily P&L of this position is approximately: $\Delta\Pi \approx \frac{1}{2}\Gamma S^2 \left[\left(\frac{\Delta S}{S}\right)^2 - \sigma_{\text{imp}}^2 \Delta t\right]$ where $\Gamma$ is the option's gamma, $S$ is the stock price, $\Delta S$ is the stock's move over the interval $\Delta t$, and $\sigma_{\text{imp}}$ is the implied volatility used to price the option. Then explain: why does a long gamma position profit when realized volatility exceeds implied volatility? What is the economic intuition behind each term?