The trick is in the substrate and amount of material necessary to achieve any kind of functioning device:
Organic semiconductors can be processed at sufficiently low temperature so that you can use flexible substrates like plastic foils or paper, even when you're doing PVD of small molecules in vacuum. To a certain extent, this also works for oxide semiconductors, but not so much for Si or other conventional materials because of the high processing temperatures.
Additionally, to manufacture a working device, you need very little material with organic semiconductors, layer stacks with a total thickness of 500nm or less are fairly realistic for OLEDs and solar cells and you probably need even less for transistors. Compare that to the average thickness of your plastic / paper substrate and you can see that the layers would stay intact even for fairly small bending radii.
Usually organic semiconductors are made of polymers or large organic molecules, which have van der waals forces along their molecular chains, while inorganic semiconductors are made of ionic compounds, which have ionic interactions. When strain is exerted, organic molecules or ions are forced apart, long chains of organic s/c can still hold them together, while ionic interactions will disminish, so the inorganic s/c are broken.