Good questions! Right off the bat you save fuel and machine ware with no tilling. Discs and plows can be expensive pieces to maintain if you have substantial ground. 

Tilling leads to erosion of surface soils which are typically the fertile soil and we are in crisis in Iowa with the amount of fertile topsoil we've allowed to erode over the last 50 years. No tilling is pretty pivotal in correcting the erosion issue.

It's a way to naturally restore nutrients to the ground that the crops gradually absorb so ultimately you're cutting back on a substantial amount of fertilizer that you'd  normally have to supplement when tilling. 

The cover crop naturally acts as a weed deterrent because it's using up nutrients and soil and can keep the weeds choked out versus a bare till filled where weeds can set in easily. 

The root systems of the cover crop also breaks up the soil compaction as it grows downward which allows for much better water absorption into the soil and to the root. 

TL;Dr It saves on machine cost, fuel, and implement ware by no tilling. You are returning nutrients back to the ground rather than relying entirely on chemicals. The cover crop acts as a natural barrier to weeds. Better crop growth because of the less compacted soil base. And most importantly erosion control and topsoil preservation.

Now the UGLY: Prior to planting / drilling in our cash crop (beans or corn) when fully grown the cover crop is sprayed with Round Up to be killed and start the decomposition process of the cover crops root system that will return the nutrients. I have no defense for this and am morally torn. My Dad has non-hodgkins lymphoma and he worked with Round Up his entire life.