Two shares, A and B, each cost exactly 100 dollars. An expert who has never been wrong tells you: There is a 50% chance that A will go up to $101, and a 50% chance that A will drop to $99. There is also a 50% chance that B will go up to $150, and a 50% chance that B will go down to $50.
If you buy A, there is an equal chance that you make one dollar, or lose one dollar. If you buy B, there is an equal chance that you make fifty dollars, or lose fifty dollars. Buying B is just a bigger bet than buying A.
Now you buy a call option which gives you the right to buy A or B for $100. If A goes up to $101, you make one dollar. If A drops down to $99, you lose nothing (because it is an option and nobody can force you to exercise the option). So there's a 50% chance to win one dollar, and a 50% chance to win or lose nothing. That option is worth 50 cents. But if you have a call option for B, there's a 50% chance you make $50, and a 50% chance you make or lose nothing. This option is worth $25.
Normally, volatility increases your chances of winning a lot or losing a lot. With call options, volatility increases your chances of winning a lot or losing nothing.
Now look at a $110 call option. There is no chance that A will reach $110, so that option for A is worthless. But there is a good chance that B will reach and exceed $110, so the option is worth a lot of money.