How much wear and tear does this impact the engine, transmission, etc? Or is it just less impact than just throttling the accelerator going uphill?

The manual says to use it in downhills as well. But I don't understand why? I'm not towing anything, so maybe that's the reason for my misunderstanding.

what does it actually do, mechanically speaking?

 

Limits you to 3rd gear as the top gear vs 4th gear.

There are situations where it is handy so you aren't lugging the engine or riding the brakes.

Not really going to hurt anything to use or not use the function.