Practically any liquid will do.

Simple answer:

Robert Bunsen told the paper to do so.

Complicated answer:

Its because the difference in refractive index between the

paper fibers and air is big (about 1.5) whereas the difference between

the indices of fat and paper fibres is about 0.1 or less.

(I'm too lazy to look up the exact figures).

When light shines on the normal paper, the beam is broken and reflected

at any boundary between fibers and air. Much of the light

comes back, thats what makes the paper appear "white".

In the stained area the boundaries beween fat and fibers do not

deflect the light, the ray goes through.

A more sphisticated version of that stain is a Chrisiansen Filter.

Georg

The oil fills in the gaps between the fibres in the paper. The paper fibres have a much higher index of refraction. The oil adhering to the cellulose fibres lowers the index of refraction of the cellulose and also fills in air voids or gaps as mentioned earlier, so that visible light passes through the bag with significantly less scattering. Meaning that the light that got reflected from the other side when the paper was dry, gets transmitted through when the paper is soaked in oil or butter.