I read this book many years ago and stuff by the Bloomsbury group in general which is where the ideas surfaced. As far as I can recall I think the central idea is that true art has a characteristic called 'significant form'. And by educating the eye this significant form can be recognised by aesthetes. It is a somewhat circular argument. If something has significant form it is art. Art has significant form. But it is difficult to define what significant form is. Significant form induces an 'aesthetic emotional' response in the viewer.
In context it was a way of educating/introducing people in England at the time to appreciate more avant garde artistic movements from France such as Fauvism and Expressionism and also to appreciate abstract art with which the English at the time were largely unfamiliar. The Bloomsbury set was largely responsible for introducing and being influenced by this type of work in Britain (see works by Duncan Grant and Vanessa Bell for example).