How does "That's It" makes fruit bars with no preservatives

A couple things - first, fruits that are dried for storage are not always dried hard - they can contain a fair bit of moisture, softness, flexibility while still being dry enough to be shelf-stable, especially if stored in a well sealed bag or container.

Also, the fruit puree or paste might have been heated in the processing, effectively pasteurizing it. If it is also stored in airtight wrappings after this step, it can be treated as if it were canned - if the bacteria present were killed, and no more allowed in by an airtight seal, the food is effectively kept bacteria free. This can be more effective if the food is stored in modified air packaging, as paparazzi's link suggests - stored without oxygen in the package, perhaps replaced with nitrogen or something stable and neutral.

If the fruit wasn't heated (there is a market for "raw" or uncooked food, and I'm not familiar with the limits to which a food can be heated before it is not longer considered raw, mild heat while dehydrating is often not considered a problem) - it still may have been sterilized by some other method before packaging - food irradiation or high pressure (pascalization). Again, if stored in an airtight container, possibly with modified air packaging, the food would stay preserved longer because there are no bacteria that can reach the food to spoil it.

I don't make bars, but I have tomatoes (a few) from 2013 that are still good, and all the local apples I'll have this year are last year's due to the spring weather this year eliminating the crop as flowers. Both of those are simply dehydrated/dried by me at home.

Commercially produced raisins work similarly, at least the dark ones, AIUI. "golden" is code for "treated with sulfur dioxide to prevent browning" and the same is seen in dried apricots and many other commercially dried fruits where browning is not a sales improvement point. Note that both of the above foods are somewhat moist, yet they keep. Sugar (from the fruit itself, in this case) can be a good thing in this context.

I would presume that a two-fruit "bar" would be made by smooshing two wet fruit types together while wet, and then drying the result, possibly with some grand mechanical wizardry to make the end product as a bar, or possibly in a mundane sheet that is mundanely sliced into bars.

If you look at full-sugar marmelade/jam, it has a reasonable chance of not spoiling quickly at room temperature. This is because of acidity, and because of the water activity being reduced by sugar - the biggest remaining spoilage risk is by mold growth. And a jam has high water content compared to dried fruit, which also contain acid and sugar, naturally or added - and less of these will be needed to sufficiently lower the pH and reduce the water activity since there is less water to dilute acid/create water activity to begin with.

I think that's it bars are just natural and have an expiration date quicker than most other bars with preservatives. They would be in jail by now if they were lying because the workers there would report them.