It's a man made pond in Florida, it's not deep at all. Definitely not something you'd swim in though, lots of funky stuff grows in warm stagnant water like that, plus ever present alligators.
The amoeba is called Naegleria Fowleri and nagelerisis is a terrifying way to die: even with treatment there's a 95% mortality rate.
The amoebas enter the nose when they traverse the olfactory nerve via something called the cribriform plate to enter the brain. Once there, they mistake neurons for bacteria, which they consume.