Boltzmann brains in the sense preferred by some authors (notably https://arxiv.org/abs/hep-th/0405270 in section C., which is the sense I'll deal with in this reply) are set in thermal equilibrium with chaotic fluctuations away from the equilibrium points that persist only briefly before equilibriating again.

Think of a gas of neutral atoms that has fallen into equilibrium warm temperature over a long, long, long period of time. Being warm the atoms bump into each other, chaotically. Very occasionally the bumps form what to humans would appear to be clusters or even higher patterns. Even more occasionally the higher patterns might monmentarily resemble something familiar and larger scale.

An analogy might be cloud-watching. Swirling changing clouds might look like birds for a little while, before looking like something different.

The Boltzmann brain is a fluctuation out of some unknown equilibrium gas into a pattern recognizable as an adult brain -- with thoughts and memories. Those memories are false ones; they did not form from actual experience, they just fluctuated into existence. The memories won't persist either, as the brain-and-memories will quickly melt back into the hot equilibrium gas.

The argument is that the entire universe as we observe it could be a fluctuation of this sort, only it takes a long time to melt. Various arguments have been presented in the past hundred-ish years which point out that without some means of suppressing smaller fluctuations, one could simply have the solar system with false shells of light falling upon our eyes and telescopes, or a momentary "Boltzmann planet" where briefly the whole Earth in some state freezes-by-fluctuation into existence before very quickly melting back into equilibrium. The Boltzmann brain is just an attempt to find the smallest such system, determine how much less complicate it is than any of the bigger systems (like a persisting universe), and wondering why Boltzmann brains aren't everywhere, including practically everywhere in a universe that fluctuated into existence.

> Matrix only larger scale

It's more like if a single instant in The Matrix was formed after a long long long period of time of filling The Matrix's memory with random values. It's closer to the https://en.wikipedia.org/wiki/Infinite_monkey_theorem -- the monkeys will make fractions of Shakespeare embedded in or peppered with (or both) strings of gibberish. Interesting fluctuations will be surrounded by (included preceded by and followed by) uninteresting ones.

> reconcile the arrow of time

There's no really good arrow of time in a system in total equilibrium. Fluctuations that are momentary do not produce a good arrow of time either. However, if we let things fluctuate into very low entropy states that take some time to decay back into equilibrium, then we can base an arrow of time on that, pointing in the direction of the second law of thermodynamics. One option is to set up some boundary around the region of low entropy that interacts with the equilibrium state outside, somewhat analogous to a cell wall, cellular membrane, or ablative heat shields on orbital re-entry vehicles. A system with a boundary is of course even lower entropy (and thus a rarer fluctuation) than a system that is subject to immediate melting back into thermal equilibrium.

> time always moves forward

Our human experience of time is that it flows in one direction, and that direction is in increasing entropy: we don't see smashed teacups reassembling and flying up onto tables, for example. We also don't see clusters of galaxies rushing towards each other as empty space vanishes. (Empty space is extremely high entropy). We also don't see black holes evaporating at this time (black holes are very high entropy).

We don't really know why there's so much lower entropy -- thanks to the metric expansion of space, and the collapse of stars and other matter into black holes -- in our past, though.

Penultimately, empty space is extremely high entropy because you can rearrange small subvolumes of a large volume of empty space around arbitrarily and it retains all the same characteristics as empty space: empty space is pretty much empty space "all the way down". The real virtually empty space between galaxy clusters expanding apart is still very very very high entropy, even though it has starlight and cosmic microwaves and so forth passing through it. Likewise, you can rearrange things within a black hole horizon pretty much arbitrarily and the outside observer will still determine the same horizon; you only change the horizon by letting things fall in, or waiting for evaporation (and if Hawking evaporation happens, the regions around black holes just emit a high-entropy blackbody spectrum anyway, rather than revealing details of how things inside the shrinking horizon are arranged).

Lastly, returning to the arrow of time: more empty space (from expansion) and more black holes (from gravitational collapse, accretion, and so on) leads to more entropy. So we can build an arrow of time from the past with less empty space & black holes to a future with more empty space and black holes. There are local pockets of low entropy visible in our sky (and on our planet), and while those may decay slowly compared to the metric expansion of space, they're already just a dwindling, diluting fraction of the cooling-and-equlibriating-as-it-expands universe.