Flex fatigue resistance defines an elastomer's structural ability to withstand repeated flexural motion from what would be understood in metalurgical circles as "Work Hardening" or "embrittlement".
Either by the addition of increased elastomeric constituants or by the nature of the polymer chain design itself, the elastomer is given a net potential to withstand being flexed back and forth, within a given temperature range, (as temperature will also affect fatigue resistance).
Consider a "living hinge" on a cheap plastic box, where the hinge is formed simply by creating a seam that might be only .010 inches thick. The more the hinge is used, the more likely it will be to fail from flexing, over time.
Polypropylene for example, has been used for years in these kinds of cheap applications in toys, automotive components and even blow molded products. Adding an "elastomer" to the base material, will icrease the flex fatigue resistance of the final product accordingly.