How Could Sheep Graze these Steep Hillsides Without Leaving any Marks?
Those lines have nothing to do with animals.

Those lines are generated by solifluction and frost heave. It is sometimes referred to as cryoplanation or soil creep. Basically as the thin hillside soil freezes and thaws it expands and contracts and because there is a slop it moves downslope. In addition rain combined with how the surface thaws before the deeped ground causes a plane of weakness and movement. It can form a variety of patterns but on strong slopes it tends to form stripes or ripples. 

That area of England is well known for such landform. Below is a picture from Dorset and another from the isle of wight. They don't only occur on grazed ground, but if ground has trees they act as anchors for the soil and you don't get the more noticeable lines and nobody is going to graze animals on ground with nothing but trees or hills with no soil which also will not have them. 

 



1. Sugden, David E. (1971). "The significance of periglacial activity on some Scottish mountains"

2. frost creep and gelifluction features a review