How does grease turn paper clear?

The grease soaks into the tiny open spaces in the paper and into the
fibers, thus creating a much more homogeneous mass - like glass -  in
which the light is not diffused as much.  It also coats the surface,
with the same effect, as does a liquid on the surface of ground glass 
(glass that diffuses light due to a rough surface produced by abrasion
or etching).

To some extent, warm wax will produce a similar effect.