A Boltzmann brain is a self-aware mind that pops into existence by sheer random fluctuation — complete with false memories of a life and world that never happened. In a large or eternal universe, such flukes are possible… and that's the problem.
The paradox
Forming a whole orderly universe by chance is astronomically unlikely. Forming a single brain with fake memories is far more likely (it needs far less coincidence). So in some cosmological models, randomly-fluctuated observers should vastly outnumber real evolved ones — meaning you are statistically more likely to be a momentary fluke than a genuine person with a real past.
If the math says most observers are flukes, then "I remember yesterday" is no evidence yesterday occurred.
Why it's taken seriously
It's used as a reductio — a theory that predicts mostly Boltzmann brains is considered broken by that very prediction. It rhymes with The Simulation Argument and The Doomsday Argument: all three use self-location ("which observer am I likely to be?") to unsettling ends.
Related: The Simulation Argument · The Doomsday Argument · The Hard Problem of Consciousness