I am an OSGS (Optical Sciences Graduate Student) and you don't need Quantum Mechanics to explain the experiment above, all you need is classical wave optics.
Linear polarization is electric field in a specified direction, lets say you have the electric field oscillating in the x direction and in the y direction for the first slit and the second slit respectively. Those directions are orthogonal to one another, so cannot interfere (the inner product is zero). But, if you have some component from both slits in some direction (for your example you will be getting out sqrt/2 of the x component in the 45 degree direction and sqrt/2 of the y component in the 45 degree direction when you insert the 45 degree polarizer, which is basically equivalent to the no polarizer case except you have reduced the amplitude). Then you have slit interference in the classical sense as illustrated here : http://astro1.panet.utoledo.edu/~lsa/_color/14_interference.htm, you will have to scroll down to see the two slit interference. Note that we see a sinusoidal pattern because our eyes view the time averaged irradiance (intensity) of the wave pattern, the the wave pattern itself.
What is different about the quantum case is that you can send, say electrons, through the slits *indivdually*, one at a time and they somehow interfere, that is what is intuitively strange.