I like to think about this problem graphically.
The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note that the option value as a function of the underlying is not linear.
Imagine the option is on 100 units of the underlying stocks, and right now, the option Delta is 0.6. The straight line is a linear function, and it depicts the value of a holding a portfolio of 60 stocks.
Let's imagine we are long the option but short the 60 stocks: therefore the option is Delta-hedged at this very point in time.
Note that when the value of the underlying stock increases, we lose money on the hedge, but we make money on the option: and because the option value is non-linear, we make more money on the option than we lose on the hedge (i.e. the option value line is above the straight line).
What if the value of the stock decreases? The value of holding 60 stocks decreases more than the value of the option: therefore we again make money (cause we are short the stocks).
We make money on larger moves up or down because being long the option means we are long convexity (i.e. gamma, i.e. we are long a pay-off that has a positive second derivative with respect to the uderlying: just think of it as a graph: if we are long a graph that has a pay-off $x^2$ and we are short a graph that has a pay-off $x$, we are long "gamma" (or convexity)).
That's the magic of convexity (or "Gamma").
Ps: for completeness, Option Gamma is just the second order derivative of the option price with respect to the underlying...
PPS: why do you lose money if there are only "small moves" in the underlying? Because it costs money to be long Gamma (this is true for any asset, including Bonds): even a delta-hedged long Gamma position will always require an initial investment to set up (i.e. in case of options, this would be the premium that we pay to buy the option): this premium will not be recovered for "small" moves in the underlying (specifically if those moves correspond to lower realized volatility than what had been priced as the implied volatility into the option).