The flex sensor you linked to uses flexible conductive ink printed on flexible base forming a resistor. It works when bent with the ink on the outside of the curve. When bent, the conductive layer is stretched and thus extends, resulting in reduced cross section (imagine stretching a rubber band). This reduced cross section and increased length results in an increased resistance, which can be measured.
All strain gauges work pretty much the same way except for the mechanical design. You must have already read this on Wikipedia.
I've also read somewhere that micro cracks are temporarily formed in the stretched conductive layer leading to a larger increase in resistance. I can't seem to find a reference to it now.