I'm not sure if this will help at all, but you are approaching a task that is taking you out of the graphic design field and into cartographic work. I'm a student majoring in Geographic Information Systems, and we use ArcGIS to spatially display data. There are other tools used to do this, like some commenters mentioned previously, but ArcGIS is the industry standard. Knowledge of the software is an entire career in itself, just as many of you spent years learning Illustrator and such. I don't know if it's possible to learn how to make a map like the one in your post without taking classes on the software. However you end up doing it, though, you need to remember a few thing. First off, the projection of the map (the mathematical response to distortion that is a result of flattening the spherical earth) influences how your data is perceived. Some projections are beneficial to how the client wants data to be visualized. For example, in the map you posted, most likely a Mercator projection, Africa's land area is heavily distorted. A general rule of thumb is that South America can fit inside Africa around three times (Africa being rather large). But in that projection, it is really small. The Mercator projection was created with naval navigation in mind, thus allowing navigators to draw straight lines across the ocean to sail. In you thematic map, or infographic, whichever you want to call it, the misrepresentation of Africa's land mass also lies to the viewer about the prevalence of religion on a global scale when compared to other continents. Second, you be better off finding software that allows tabular entry of data that is what the circle's areas are based on. Finally, you can take the map that is published by whatever software you use and throw it into illustrator or photoshop to finalize it. There are some free alternatives to ArcGIS out there.

Here's a slideshow about projections https://training.fws.gov/courses/csp/csp7203/resources/Map_Projections_2013.pdf