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.