First, I am not an expert at the greased line method. I'm only going on my first hand experience of actually "getting it" for the first time.
The intention of greased line presentation is to keep the fly swimming just below the surface of the water, broadside without disturbing the presentation of the fly with unnecessary mending.
In low, clear water, using small sparse flies, with a floating head, the greased line presentation is a highly effective way to fish for steelhead, so I just found out the other day. The way it works, you cast your line across and at a slight downward angle, I would call this a 45 degree angle to the currant of the river. Make a very slight mend if necessary, as the fly line straightens and comes under tension, the rod leads the fly line, don't force it or drag the fly. This method gives the fish a side view so that they take the fly from behind, according to Trey Combs.
I found myself unintentionally fishing this style the other day. It was the best day, getting hookups, and watching Marty get grabs, hook-ups, and land steelhead.
Read more about greased line presentation in "Steelhead Fly Fishing" by Trey Comb.
