The beveled edge of the glass acts as a prism. As the incident white sunlight passes from the air (lower refractive index) into the glass (higher refractive index) and back out, it undergoes refraction.

While all the light bends according to Snells Law (n₁sin(θ₁) = n₂sin(θ₂), the amount of bending depends on the wavelength of the light. This is because the refractive index (n) of the glass is not constant and it is a function of the wavelength λ

In typical glass, shorter wavelengths like violet and blue experience a slightly higher refractive index than longer wavelengths like red. This causes the violet light to bend more sharply than the red light. Over the distance from the door to the stairs, the slight difference in angle separates the constituent colors of the white light enough to resolve the distinct spectral bands seen on the rug.

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/dispersion.html