How to post-process an image to turn trees blue and a face yellow?
There are multiple things going on between the before and after image.  The first aspect is a color space conversion.  The original image is recorded in a "log" color space.  

A log, or logarithmic, color space is designed to capture more detail in the shadows and bright areas of an image allowing for the image to be manipulated more easily in post production.  While RAW files can be used in digital still photography, the size is fairly limiting for video usage outside of very high end setups (and even then it is costly due to the tremendous data rates).  To provide a mix of flexibility between cost and post production adjustment, log color spaces sacrifice data about the midtones of an image to be able to capture more detail in the areas of contrast.  This gives much greater flexibility in adjusting the color and exposure of an image in post without needing pure raw data, however it makes the images take on a decidedly flat appearance as it is not a color space designed to emulate human vision, but rather to capture information about the image that can be used to produce an image in a linear color space that will look more natural to the eye.

The second thing being done is a color grade pass.  Color grading is the process by which a colorist adjusts the color of a video and gets a consistent look and feel between clips and to set the look of the overall video.  A variety of tools can be used, including color balance adjustments such as RGB curves which will impact the response of different colors between various different portions of the image (based on luminosity or brightness).  Hue adjustment tools, which will adjust the hue (or color) of different colors within the image, or LUTs (look up tables) which provide a fixed set of color replacements that can be applied to an image to produce a new set of color values given a set of colors taken in.

My best guess at this particular before and after is that the log color was balanced and adjusted to a normal linear color space and then a filmic LUT was applied that has a bluish tint to the greens.  It may have been an out of the box LUT from a vendor or a custom designed LUT depending on the colorist.  It is certainly attempting to emulate a more filmic look though to emulate the color response of film more so than typical video.