1) When light falls on paper, a part of it is transmitted and a part is scattered, a part is absorbed and a part is reflected.
2) The degree of transparency or opaqueness of a medium will depends on the ratio of light transmitted, versus light reflected or scattered.
3) Higher the light transmitted and lesser is reflected or scattered, more will be transparency.
4) The amount of scattering depends on size of fibres or fillers and the difference in the index of refraction between the particles and the surrounding medium.
5) The paper fibres have higher index of refraction probably much greater than 1.5.
6) The oil or fat also has a high index of refraction. So that it nearly matches the index or refraction of the paper fibres and it reduces the scattering significantly.
7) The fat adhering to the cellulose fibers lowers the index of refraction of the cellulose and also fills in air voids, so that visible light passes through the bag with significantly less scattering.
8) The oil connects the fibres in the paper with a liquid which can transmit by refraction (rather than scatter) light that falls upon it. As a result, the paper stained with oil is turned transparent.