Instead of thinking of it as a cassette or gear, think of it as a lever. A larger gear has a longer span, or lever, from the axle to the outer edge of the gear where the chain pulls on it, so when you pedal, the chain is, in effect, pulling on a longer lever which gives you more power for the hills but less speed.

Of course the opposite is true for the smaller gears on the cassette, there you are pulling a shorter lever which gives you less mechanical advantage power wise for hills, but the ability for more speed.