Why does fat make chapati/parathas/pooris remain soft?
The "moistness" of the bread - the soft texture - is actually from oils and fats as well as water - grain has some of these naturally, especially whole grain. Adding in extra fat, such as butter, shortening or cooking oil, makes the baked good seem extra moist and soft - here is a good overview of baking, and the role of various ingredients in dough. Fats are a tenderizer when added to a dough - it makes the baked good softer, which we associate with "moist." 

Fats and oils also evaporate much more slowly than water, and prevents water from re-absorbing into the bread, keeping the bread from becoming brittle and hard and off-tasting (stale) for longer. 

Without extra fat, the bread relies on the oils and protein found in grain, and the water content of the bread. This water will eventually evaporate, leaving the bread brittle - and worse, it will slowly re-absorb some moisture from the air, and this can lead to a dry, mealy, chewy texture and unpleasant flavor - staleness, in other words. (This is why croutons should be baked for best flavor rather than just left to go stale.) The natural oils and proteins will likewise firm up and stiffen as the bread cools.

For leavened flatbreads, preferment is a common technique to extend shelf life (and improve flavor) - this method of leavening does a good job of locking water-moisture into the bread without requiring extra fat or other preservatives. Here is a method of prefermenting naan.

For unleavened bread, Mexican cuisine is where you should look for techniques involving storage of whole-grain flour flat breads - in some regions, flour tortillas are eaten with every meal, but generally only prepared once a day.


Air-tight storage is key. A zip-close plastic bag with a cloth to absorb excess moisture is ideal.
Re-heat the bread in a microwave. It will soften the proteins and oils, without drying out the water content the way an oven or toaster will.