[H]ow can it be that natural selection "favors" a particular coloration during winter and summer?
Camouflage for example. In regions, where it snows some animals would take advantage to have a fur colour that match the substrate in which they live and they may therefore be selected for having a white fur in winter and a non-white fur in summer.
Consider for example Lepus lepus in summer
vs in winter
How can it be proved, that these animals change their coloration because it is useful for them?
There are a whole lot of possible experimental design.
Considering seasonal coat colour in response to predation, a simple and obvious approach would be to experimentally dye the individuals and see how their survival varies depending on the colour they have been dyed.
Sometimes, you may have a sister clade that is non-plastic which it might be of interest to test against (depending on the exact question at play).
I'm more interested in ecological impact on coloration [..] I'm looking for the articles providing evidence
Cases of local adaptation (such as Hoesktra 2006 for example) or other evidence of natural selection on coat colour such as differential selection among lineages are probably way more common. But if you are specifically interested in seasonal variation then, here are a few papers I could find with a quick google scholar search showing selection on seasonal variation for fur colour.