Hi! The map is totally independent from BioGeoBEARS. You can do such maps in R, with packages like maptools, sp, and rgdal. The world map dataset is called "wrld_simpl". However, coloring custom sections within that map requires making new shapefiles, this can be done but requires some GIS experience or a lot of googling. For a one-off, it is probably easier to do it by hand in a graphics program.
For the BioGeoBEARS part, that looks like a phylogeny with per-area probabilities plotted, see eg:
I would do the phylogeny & area colors first, then at the end make your map with colors to match.
Cheers!
Nick