The best I can tell from my research on the web and interviewing pet owners, it has something to do with helping to settle their stomachs. Opinions vary as to whether grass is meant to actually calm their grumbling tummies or cause them to sit up whatever is ailing them.
For me, and others that I know, it's all about yacking things up. I cannot say definitively whether it's because grass tickles their already sensitive stomachs or there is something on the surface (ie, fertilizer, etc.) that might be the cause. I'd recommend using caution and not letting them chomp away without restraint in particular in areas that you don't know how they tend to to their lawn.
In 90% of the times that my senior pal decides to dine on the lawn, he ends up spitting up exactly the same amount that he ate. This happens every few weeks and at least in my case it's because I fed him early the night before and we went out before eating that day.
This is not a recommendation to feed a dog and immediately take them out for exercise. Dogs who get out too soon after eating, in particular larger chested dogs, are more likely to develop bloat which is both uncomfortable and dangerous. Wait at least one hour after a meal to go outside to be safe.
Related articles:
Why Do Dogs Eat Grass?
How to Feed Dogs at Risk for GDV
