Gamma tells you how much your delta changes as the underlying price moves back and forth. If the gamma is positive it means that the delta increases as the underlying goes up, and delta decreases as it goes down. I.E you get longer the rising asset, and shorter the falling asset. This is ideal as gamma acts as an auto-(de)leverager.
Let's take a real example. The asset price is 1, with 30 DTE options trading on it. We select the option with the largest gamma, which is right around the spot price, and it has a volatility of 20%. This equates to a Black Scholes gamma of: $$\frac{n(d1)}{s \sigma \sqrt{t}} \approx \frac{1}{1 * 0.2\sqrt{2\pi \,t}} \approx 7$$
With our initial $\Delta$ at 0.51, this means that for any change in the underlying price, $dS$, we expect our delta to change by $\Gamma * dS = 7*dS$. So if the underlying moved to 1.01, our new $\Delta$ would be expected to be $0.51 \, + 7*(1.01-1) = 0.58$, evaluating the delta under BSM shows that our new delta is 0.5801, up 0.691 from 0.511, it is less than 0.7 because of higher order greeks which change the gamma as price moves. Remember that greeks just describe risk, or sensitivities, they are slopes of P/Ls through different input values like spot:delta, IV:vega, and time:Theta. Gamma is the slope of delta with spot, speed is the slope of gamma with spot, and so on. Incorporating greeks into risk analysis tells us our local sensitivities to input factors.
When looking at factors like delta, it is intuitive to say how much money we will make or lose. For example, with a delta of 0.5, we expect to make 0.5 USD for every 1 USD move in the underlying. So our P/L can be written as: $$ PnL = \Delta * dS = \frac{dO}{dS} * dS$$ We try and model dS so that we understand our dollar risks with current positions. For higher order greeks it is less intuitive, but I think this derivation is quite nice. Consider your position with$S = S_0$. As S moves to $S_1$, the P/L between gained over the two points is given by the average $\Delta$ between $S_0$ and $S_1$, multiplied by $dS$: $$ PnL = \frac{\Delta_0 + \Delta_1}{2} * dS$$ We know that $$ \Delta_1 = \Delta_0 + \Gamma_0 * dS $$ $$\therefore PnL = \frac{\Delta_0 + \Delta_0 + \Gamma_0 * dS}{2} * dS = \frac{2\Delta_0 + \Gamma_0 * dS}{2} * dS = \Delta_0 * dS + \frac{\Gamma_0}{2} * dS^2$$ In the case of delta hedged portfolio at $t_0$, the offsetting delta term $-\Delta_0$ leaves the PnL as $$ PnL = \frac{\Gamma_0}{2} * dS^2 $$ The dS^2 term is related to the variance (and thus volatility) of the underlying, consider the starting model that we use for GBM: $$ dS = \mu S \, dt + \sigma S \, dW$$ Taking expectation of the square of dS leaves us with: $$ dS^2 = \sigma^2 S^2 \, dt $$ Leaving our PnL formula as: $$ \frac{\Gamma_0}{2} * \sigma^2 S^2 \, dt $$ However as you know already, gamma is not free, one must pay theta away. Without even deriving a formula for theta, we know that for a fair market, the expectation of owning an option should be 0. As in nobody expects to make or lose money when transacting (of course that's different due to risk premiums, tx costs, etc, but fine for this example). If our instantaneous P/L from $\Gamma$ over the period $dt$ is $x$, then our theta must be $-x$.
One interesting point to consider is that the price of the option must be equal to the expected profits for holding it over it's lifetime. For a market neutral position (instantaneously riskless) that is kept neutral over it's life through continuous hedging (without cost), the price of the option should be equal to integral (continuous sum) of $\Gamma$ PnL over it's life. Formulaically: $$ Cost = \frac{1}{2} \int_0^T S_t^2 \sigma_r^2 \Gamma dt $$ Assuming $S_t$ as a constant, which, for small $\sigma_r$, and 0 carry is not too far from truth, the formula for an ATM option is approximately: $$ \Gamma_{s,v,t} = \frac{1}{S * \sigma\sqrt{2\pi \,t}}$$ which simplifies the cost formula to: $$ \frac{1}{2} \int \frac{S \sigma_r}{\sqrt{2\pi \,t}}dt = \frac{S \sigma_r \sqrt{T}}{\sqrt{2\pi}} $$ So now hopefully you can see the relationship between gamma, realised volatility, and the intimate relationship between those and the price of the option. Finally, since we pay $\Theta$ away at a proportional level to $\frac{\Gamma S^2 \sigma^2}{2}$ By buying or selling options and hedging them to maturity, we are replicating them at realised volatility, whilst paying implied volatility, this means that our final payoffs are given by $$ \int_0^T \frac{\Gamma S^2}{2} * (\sigma_r^2 - \sigma_i^2) dt$$
