Does temperature affect the index of refraction? I would like to know the physical reasoning behind the fact that temperature indeed does change the index of refraction of a certain medium.
(Is there an easy experiment for me to try this at home?)
 A: Refractive index does not change because of a change in temperature, it changes because of a change in density. Increased temperature usually reduces density because most materials expand when heated (but some contract). If density is kept the same while temperature changes (eg by varying applied pressure) then there is no change in refractive index.
A simple Science Fair experiment which you could do at home/school is described in the following reports :
http://cssf.usc.edu/History/2008/Projects/J1625.pdf
http://cssf.usc.edu/History/2006/Projects/S1513.pdf
However, these experiments do not make any compensation for the change in density, so it is impossible to tell if the results obtained are due to change in temperature or change in density. 
It is difficult to eliminate or compensate for changes in density during the experiment. You could compensate after the experiment by obtaining and plotting data for density of the material at various temperatures within the experimental range and comparing with your measured variation in refractive index. If the two values remain in proportion as temperature varies then you can deduce that the variation is entirely due to density and not at all due to temperature.    
A: The change of density with temperature is not the only source of temperature dependence of refractive indices. Another cause is the change of polarizability with temperature (at constant density) - http://photonics.intec.ugent.be/education/IVPV/res_handbook/v2ch33.pdf, equation (35)
A: Light is an electromagnetic radiation having both electrical and magnetic fields . It’s motion through a matter has its interactions with it which depends on the matter through which it has to pass. The molecules themselves are moving with Brownian motion as in liquid or vibrating about its position , a property related to its energy state , which depends on temperature and molecular interactions with electromagnetic wave depends on electron clouds and it’s fields, it encounters while passing through it.
In case of gases, the void space increases with temperature or in layman’s language the density decreases so the hindrances to the passage of light decreases means refractive index of the medium decreases.
Let’s us have some optically active medium and you are able to control the concentration of optically active reagent in liquid, then not only light bends but it’s plain gets rotated depending on the optical reagent concentrations.
Here reason is electromagnetic interactions with optically active medium which not only slows down light but it bends too depending upon the intensity of such interaction on the path of plain polarised light. Here you observe bending ( change in refraction ) and Rotion of plain( due to optical interactions ) .
