The fungi infection is a plague that spread across the Mushroom Kingdom and infected most, if not all, of its citizens.
Little is known about the early stages of the fungi invasion. Peach, a living cordyceps and harbinger of the fungi, overthrew the reigning government and infected any remaining citizens. Those who survived escaped to the neighboring kingdom.
Koopa Kingdom's Involvement
The refugees, known as Goombas, escaped to the nearby Koopa Kingdom, which was most likely sealed off from the Mushroom Kingdom. Pleading for help, Bowser needed to figure out a way to help the Goombas without being spread to neighboring kingdoms and getting rid of the Goombas.
Bowser repurposed the infected civilians as rocks, bricks, and horsetail plants, that last one to stop the growth of anymore fungi. He also took the cordyceps, known as Toadstool or Peach, and brought her back to the castle for experimentation.
However, a servant known as Mario attempted to rescue the "princess." Bowser attempted to stop him, unfortunately, Mario rescued the cordyceps. Bowser has continued to try and save the Mushroom Kingdom, but all efforts have been fruitless.
Involvement of Others
Aliens and other kingdoms have attempted to stop the spread, only to fail. Sarassland may very well be under its spell, and Yoshi's Island may be infected.
Mushroom Plague's State Today
As of March 7, 2015, the hosts seems to have become smarter and create machines, and have learned to save their own kind, as seen in "Captain Toad: Treasure Tracker." Mario seems to be one of the few things standing in the way of eradicating the fungus.
It has, however, spread across a large portion of the U.S.A., to the point where some think the whole world is infected. The Last of Us, for example. Luckily, there seems to be someone immune to the fungus, known as Ellie.