Structure of diamond
Where I am struck is that I knew carbons occupy zinc sulfide like
  lattice so, how can this line be true.


Zinc sulfide can exist in two different crystalline forms, Zincblende (sphalerite) and wurtzite.  Zincblende is face-centered cubic (also known as diamond cubic), each ion is tetracoordinate and has local tetrahedral geometry - just like the carbons in diamond.  The wurtzite structure is hexagonal close-packed with interconnected 6-membered rings.  You may have been thinking of the wurtzite structure, while it is the zincblende structure that is related to diamond.  You can see pictures of the two structures here.