First up, don't get too excited, when they say flexible displays it doesn't really mean flexible displays in the sense we all want it to, it simply means the displays on these upcoming devices will be curved and utilize flexible technology in the displays to make them more resilient and less prone to breaking. Flex displays will also allow further reduction in the size of bezels. Translating from Korean is not my forte however, so there could be some minor flexibility in the new devices, but the internal components won't be so there's obviously a limit.
Samsung has drip fed us some information about what could be the limited edition, flex-displayed Note 3 Active and have confirmed they will announce a new device next week for an October release. If it is the Note 3 Active, expect the same specs as the original along with other changes, like dust and water resistance, which will also boost the price tag. If you worry about breaking the screen of your smartphone this is undoubtedly good news for you. A GFXBench score for a device codenamed SM-G910S, which everyone thinks is the Note 3 Active, has been poo-pooed by some as a new device range entirely, so let's wait and see what the announcement reveals.
As for LG, the new Z series, which might also be called the Z1 or the G Flex, will be launched in October also, but when we'll see an official launch date we don't yet know. There's not much in the way of specs on this device either yet, other than featuring a Snapdragon 800 chipset, Android 4.3, 6-inch display and LTE-A support, so the details will remain blurry for now. As far as I can glean from the Korean website on which the news appeared, the LG Z will have a banana-like curved screen with minor screen flexibility and the screen will, according to LG, be near-indestructible. No matter what the final form factor it'll be quite the novelty and sooner or later we'll see more flexibility creeping into smartphones as internal components adjust.