The basic idea behind Boltsmann brains is that if intelligence arose randomly in the universe, it's vastly more likely to arise as a lone speck of order in an "entropic soup" than to arise in the middle of a huge bubble of order like we observe around us. Most significantly, it's an approach for refuting the selection bias argument against the anthropic principle.
Boltzmann proposed that we and our observed low-entropy world are a random fluctuation in a higher-entropy universe. Even in a near-equilibrium state, there will be stochastic fluctuations in the level of entropy. The most common fluctuations will be relatively small, resulting in only small amounts of organization, while larger fluctuations and their resulting greater levels of organization will be comparatively more rare. Large fluctuations would be almost inconceivably rare, but are made possible by the enormous size of the universe and by the idea that if we are the results of a fluctuation, there is a "selection bias": we observe this very unlikely universe because the unlikely conditions are necessary for us to be here, an expression of the anthropic principle.
The Boltzmann brain paradox is that any observers (self-aware brains with memories like we have, which includes our brains) are therefore far more likely to be Boltzmann brains than evolved brains, thereby at the same time also refuting the selection-bias argument. If our current level of organization, having many self-aware entities, is a result of a random fluctuation, it is much less likely than a level of organization which only creates stand-alone self-aware entities. For every universe with the level of organization we see, there should be an enormous number of lone Boltzmann brains floating around in unorganized environments. In an infinite universe, the number of self-aware brains that spontaneously randomly form out of the chaos, complete with false memories of a life like ours, should vastly outnumber the real brains evolved from an inconceivably rare local fluctuation the size of the observable universe.