I gathered this from a talk by Sean Carroll that I attended, and it was supposed to be a consequence of the standard picture. All the Boltzmann brains come up in the way distant future, after thermal equilibrium, as random fluctuations. Carroll regarded this as a defect of the normal approach, and used this as a launching point to speculate about a different model.

I wish I had a more precise reference, but this isn't my area and I only heard this one talk. But I think this issue is discussed in his book From Eternity to Here. Here's a blogpost that, I believe, faithfully summarizes the relevant part of the talk. The normal solution to Boltzmann brains is to add a past hypothesis. Here is the key part where the post discusses the benefits and shortcomings of this approach:

Solution: Albert adds a Past Hypothesis (PAST), which says roughly that the universe started in very low entropy state (much lower than this one). So the objective probability that this is the lowest entropy state of the universe is 0—meaning we can’t be Boltzmann brains. As a bonus, we get an explanation of the direction of time, why ice cubes melt, why we can cause things to happen in the future and not the past, and how we have records of the past and not the future: all these things get a very high objective probability.

But (Sean Carroll argues) this moves too fast: just adding the past hypothesis allows the universe to eventually reach thermal equilibrium. Once that happens (in about 10100 years) there will be an extremely long period (~10^10120 years) during which random fluctuations bring about all sorts of things, including our old enemies, Boltzmann brains. And there will be a lot of them. And some of them will have the same experiences we do.

The years there are missing some carats. Should be 10^100 and 10^10^120.

This is always hard with infinities. But I think it can be a mistake to worry about this too much.

A rough way of making the point would be this. Pick a freaking huge number of years, like 3^^^3. Look at our universe after it has been around for that many years. You can be pretty damn sure that most of the beings with evidence like yours are Botlzmann brains on the model in question.