The second law of thermodynamics states that the entropy or disorder of a system always increases over time. A hot cup of tea, for example, represents an ordered state, since the atoms in the teacup have significantly more energy than its surrounding. The second law tells us that the teacup will cool down, meaning the energy will be dispersed over its surroundings making the entire system less ordered.
After this law was established in the late 19th century, people started to apply it to the Universe as a whole and ask, why is the Universe in such an ordered state? The Universe mainly consists of stars, planets, and galaxies, separated by huge expanses of empty space. So the Universe represents a very ordered state. Given that the Universe was considered to be eternal the second law of thermodynamics would suggest that we should live in a very disordered Universe, which would not allow life to form.
There are two solutions to this problem. One version is that the Universe is not eternal and it started with very low entropy. Why the Universe started in a low entropy state is unclear and remains a key question in cosmology. The second solution makes use of the fact that the laws of thermodynamics are statistical laws and hence while the entropy should always increase over time, there is a small (but non-zero) probability that it decreases. One could use this to argue that the Universe just happened to randomly fluctuate to a low entropy state. For example, atoms could randomly assemble in a tiny region of space, which would give rise to the BigBang and create the Universe we observe today. Even though such a fluctuation is extremely unlikely, one can invoke the anthropic principle, which states that we can only live in parts of the eternal Universe which is subject to such an unlikely fluctuation.
Given that the probabilities for such thermal fluctuations decrease exponentially with the number of atoms involved, it is far more likely to have small fluctuations in small regions of the Universe compared to creating our observed Universe.
The Boltzmann brain is a thought experiment in which such a random fluctuation would form a fully functional brain without any life-supporting environment. It is vastly more likely that a thermal fluctuation creates a Boltzmann brain rather than an entire Universe which can harbor life, meaning that if life really is formed by thermal fluctuations, most of the life in the Universe consists of Boltzmann brains.
Boltzmann brains are linked to the measure problem in cosmology, where one often has to deal with infinities in time and space, which means that even small probabilities can become feasible given enough time or space. If such scenarios would result in Boltzmann brains as the most common life form, this is usually considered to be an indication that this is not a viable scenario. However, the question why we live in our Universe rather than as isolated brains floating in the Universe is not really solved yet (here is a recent paper on the subject https://arxiv.org/pdf/1702.00850.pdf).
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best
Florian
