Ripping is just running a subsoil plough across the landscape (Yeoman's developed his own especially for Keyline, with specifically angled points). Ripping sounds harsher than it is as the shanks (onto which the points are mounted) are preceded by a coulter (like a large pizza cutter) that slices open the soil to facilitate easily passing of the shanks. The soil isn't turned, just lifted a little, leaving slices cut in the pasture, into which water and plant roots can penetrate. It's most often used to break up pans and facilitate deepening of the topsoil.
These ploughs often have other elements added, such as seeders & feeders, some getting the label 'super plough'.
Usually the first cut is made from the keypoint in the valley, out along the contour onto to ridge. Subsequent passes are made parallel to this (at equal offset), the outcome of which is that each the following cuts all run slightly downhill from the valley to the ridge, moving water back where it is most needed.
By ploughing just an inch or two into the subsoil, the topsoil will deepen. You'd then return after a period and plough again at a slightly deeper level, each time encouraging roots & life to penetrate further. This image will help show that:
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