Fireproof safes hold a lot of moisture so that when the safe is exposed to fire, the boiling off of the retained water keeps the items inside protected from extreme heat (until the water boils away). A downside is that permanent extra water is that it keeps that contents of the safe permanently damp (unless the safe is frequently aired out).
That is, they use the latent heat of water to buy some time in a fire, hopefully so the safe can outlast the fire. As is probably obvious, all "fire proof" safes are only fire resistant, and some say that this boiling of the water is the primary factor in their time limit, and some more accurately label them as "fire resistant" (because you can't fight thermal equilibrium forever).
So, hopefully, this makes the answer to the question clear: "Why do fireproof safes “capture and hold in moisture”?" It is because the contents of these safes are surrounded by a moisture holding material, which over time is in thermodynamic equilibrium with the interior of the safe, raising the humidity to much higher than normal relative humidity and more extreme and consistent than normal condensation. In general, the water-holding material forms a shell around the safe, and if the interior were completely water tight the contents could remain dry, but in practice, this isn't the case (see item 6 in final paragraph for a possible reason).
I know this from the documents and experience with safes I have used, just as a normal user, but I was interested to find out a bit more (eg, I've always wondered with the musty safes, "can they really not do better than this?").
That this approach is used in modern safes, is clear, for example, from this statement:
"There is a good reason for this. In order for a fire-safe to successfully resist fire, the special composite material sandwiched between the outer and inner steel walls gives off moisture. This creates steam inside of the safe."
But I wanted to know more about this, especially, how is this water held around the contents, and what is the nature of these "special composite materials", so to find that (without a "handbook of fireproof safe design"), I searched for old and relevant patents, and here are some interesting facts:
This concept was in use before this patent (pdf) written in 1868.
People also used pipes around the safe to hold the water, but that had it's own problems, as described in this patent (pdf)
Here is an example (pdf) from 1869 using plaster of paris to hold the water. I am happy to stop with this one example, but I also presume that there are more modern "special composite materials" for holding moisture.
Finally, a note on condensation: The other answer here and multiple comments bring this up so I'll address it directly and explain why it's not relevant, in my opinion. 1) By "condensation" I assume people mean that when the safe is opened and subsequently shut, humid air from outside the safe is trapped within, and then can cool down and release its moisture. 2) It's important to note that this is a small amount of moisture, at most a few grams, and would be easily managed in the normal way, which is by using a desiccant. 3) Yet, a desiccant pack won't work in a fireproof safe. 4) This condensation problem would not be any worse in a fireproof safe than in a regular safe or a sealed box. Yet the moisture problem is much discussed for fireproof safes and not for regular safes or boxes -- it's not that it doesn't exist for boxes but that everybody knows the solution for boxes (ie, desiccants). 5) It's also worth noting that paper is quite absorbent itself, and a box full of paper should act as its own desiccant, and can absorb a reasonable fraction of its own weight in water without issue, so more relevant is the ambient humidity (an infinite source of moisture), not condensation (a small source of moisture). (But this isn't relevant for gun safes). 6) People suggest that fireproof safes could be air tight, but this would risk getting into a vapor lock situation where if the safe was opened when the safe was colder than the air, then on closing the air would cool down inside the safe lowering the pressure, making the safe truly impossible to open other than by heating. (That is, you can't pry up the corner of a safe like you would a box, you'd need to fight against the pressure difference over the entire door.)