It's supposed to be second material on your mesh because there is no surface, only fur is rendered. I made it cutout and moved to surface shader from frag-vert so unity can lit fur properly, also fog now works. But cutout shader looks pretty weird without antialiasing Also it has more perfomance overhead than the original shader.
If you want this shader to be transparent: go to FurShaderSurface and for every pass change alphatest:_Cutoff to alpha. It still will be lit, but won't get any shadows
