How is downsampling and low-pass filtering done in MS-SSIM?
I assume the question is specifically the decimate by 2 as in this figure:



Decimation is the combination of a low pass filter with a downsampler. The downsampling by 2 designated by the block that has a 2 with a down-arrow is simply done by removing every other sample. This would fold in frequency everything in the upper part of the spectrum so an anti-alias filter is required prior to the down-sampling. This is no different from the requirement for an anti-alias filter prior to an A/D converter. So the ideal filter L would be a low pass filter that would pass everything that would be in the first Nyquist zone defined by the new sampling rate after the downsamping, while completely rejecting all the high frequency components that would otherwise fold in. (Such an ideal filter cannot be realized, but we approach this trading complexity with allowable distortion). 

This graphic I have below demonstrates this with a decimate by 4 for a real signal with the first Nyquist zone extending from $0$ to $F_s/2$, where $F_s$ is the sampling rate (and $-F_s/2$ to $+F_s/2$ for complex signals).   



With digital sampling (or resampling as in this case) all the spectrum centered on $mF_s$ is aliased to be centered on $0$ in the first Nyquist zone where $m$ is any integer, so here in this example the new sampling rate is $1/4$ of what the original sampling rate, thus we see all the images that are at multiples of this new sampling rate that were part of the original digital spectrum that was at the higher rate.

In this case a multi-band rejection filter is a good solution to minimize resources. Alternate approaches that are also quite efficient are Cascade-Integrator-Comb structures (CIC) and polyphase filters which are detailed in other posts on this site.