Their origin is a little mixed. When the Rift occurred, it was localized in 4 portals across the world, with a portal to the earth realm opening up practically in the middle of the Northern Valcorian Empire, a portal to the fire realm in the middle of the Cythra continent, a portal to the water realm close to the Ruzhai continent, and the wind realm opened on the north-eastern coast of Luong-Xi.
So, when these portals opened, you didn't need to be in the vicinity to know something was up. About 0.1% of the world's human and elven population seemed to spontaneously manifest what appeared to be elemental powers that transformed their bodies to 1 of 4 elemental sources. There are many theories why but nothing that's been grounded in experimental fact. Due to much fear at the time, especially since the arcane wasn't nearly as well known as it is "currently", these individuals were rounded up and isolated from the general population. At least that's what was done in the most lawful societies. Today the events relating to those poor souls is often, and shamefully, referred to as the purge.
Out of the portals flowed elemental beings apparently keen on conquest. When the time of the Rift came to an end and the portals were closed, some beings found themselves cut off from retreat, and in other circumstances such as with the tieflings, an accord was reached with remaining elemental denizens which resulted in cooperative measures.
Thus Genasi have continued to survive into today. Often they are descendants of those who survived the purge, but for many they are offspring of elemental denizens who have made Yeohven their new home. In more rural areas there is superstition that human or elven children who reach puberty run the risk of transforming into a Genasi, which has led to some very… peculiar traditions, but academics haven't substantiated that this is a genuine condition to watch out for.
In general, Genasi are found in the Tiefling empire where they are treated as proper full citizens when compared to elsewhere, but enough time has passed that Genasi are generally seen as any other race would be, but with the social and cultural baggage that few others generally cope with.
As a note on this, fire and earth Genasi are most common in the starting area, as that's where those portals were particularly. Air and water exist as well, but in fewer numbers.