When I was learning how to drive a manual, my Dad was a huge fan of them and all of our family cars were manual. I was taught that it makes a huge difference when you're driving in the mountains to use engine braking instead of your brakes. He also explained the difference in braking in traffic and using engine braking to do so and how it supposedly saves on gas as well - something about idling the engine versus continuing to have the engine running?
I'd be interested to hear from people who regularly drive in the mountains and the advantages/disadvantages of engine braking versus using your brakes and if that was some myth or actually real.
While you're accelerating, you're giving the engine gas to keep it moving and prevent a stall. While braking you aren't accelerating, but you still need the engine to keep going at speed.
If you depress the clutch, it disconnects the engine from the wheels. The engine will then get a trickle of gas (even if you don't accelerate) to keep it moving while the car breaks.
If you don't depress the clutch, the engine stays in connection with the wheels. Their motion drives the engine - the reverse of normal - keeping it moving without burning any gasoline at all.
It's not generally a big difference, but if you're in the hills of Colorado you might care more.
As for wear on the engine: the engine is actually designed to mesh with the transmission. Engine braking is not much different than accelerating.