I want to create something of that similar effect...I don't know what this style is called, I don't know what tool to use, please help me.
R (the letter R) is free software environment for statistical computing and graphics. It can be extended by means of packages that the R-core team and team and community create. For creating a video such as the above, you would want the data that describes the date of a nuclear event, the yield of the event, some metric of its decay function (perhaps the half-life of some particular isotope present in the fallout of the event), the geolocation of the event, and the country responsible for the event. You could then feed this data into a package such as animation combined with rworldmap.
R is a great way to bring data to life, visually. With the above, you can create animated GIFs, active HTML pages, or full-fledged video files that can be further integrated into digital production environments.