TLDR: The colors are recorded by the MODIS sensors on the Terra and Aqua satellites. This is an average over 20 years of data.
A few caveats:
What the satellites record is raw reflectance. This is in a certain spectral band (like red, green and blue) how much of the suns light is reflected off a single area of land from 0-100%. Our eyes are also sensors which are not at all compatible with scientific sensors used on satellites. So to get from actual red, green and blue reflectance values to colors that we can display on a computer screen and which are close to what a human would see if they were shot into orbit, we need to use a conversion formula.
I used the formula which was used to compute NASA's Blue Marble composites. However there's more scientific approaches which would actually map the spectral response of the MODIS sensor to the CIE XYZ color space, which would then allow for a pretty exact conversion to what humans would see (If you love details, see this article for more info). That takes a lot of work though so I leave it up to a motivated reader.
To get the data I present here we need to also have a map of the entire earth for what land cover is where (and when!). I've used MCD12Q1.061, in particular the Annual International Geosphere-Biosphere Programme (IGBP) classification. This is a classification derived from the raw satellite data that I also used to collect the colors.
This classification and the reflectance data I'm using is at 500m resolution, so everything gets a bit muddled. That's why water areas and urban areas look the way they do. They are mixed up with lots of other stuff. To add to it the Land Cover classification is anything but perfect, loads of areas are misclassified but on a global scope the hope is that it all evens out somehow.
I didn't manage to get data for Antarctica. Sorry. Spatial geometries often don't like crossing poles or the date line. For Antarctica they do both and I couldn't be bothered to handle those issues just to get 50 shades of white.
Also this is only land areas, basically everything which falls into the country borders which are made available by geoBoundaries. This also means that disputed border regions are removed. However all of the raw data for the border regions is available. So if you want to look at Western Sahara for example be my guest.
I left in snow. For many countries and the global averages this is really noticeable, I don't know if it was the best decision, but I think it makes sense. If you're looking at Canada in winter everything would be snowy. So the average forest color for Canada should reflect that.
Computing averages over 20 years for the entire world isn't an easy feat. To do it I used Google Earth Engine, which is free for scientific and non-commercial use. At this scale it's also almost inevitable that I made some mistakes, if you find any, please do leave an issue or a pull request on github.