I've spoken to the CEO a couple times. They've contracted some prominent French cryptographers to vet their cryptosystem, but no security analysis or whitepaper has been released.
Their KFHE scheme is different than previous schemes because it uses a different hardness assumption that leads to better performance and smaller keys. Previous FHE schemes used hardness assumptions based on ideal lattices, but their scheme uses a different assumption based on the hardness of solving a system of quadratic equations in many variables. Whether their construction truly is a practical general-purpose FHE scheme is not clear, since perf numbers for general circuits are not available (as far as I know).
Schemes based on solving systems of multivariate quadratic equations have some nice properties, including the aforementioned performance benefits as well as conjectured resistance to quantum attacks, but have proven extremely difficult to get right. Many schemes based on these assumptions have been severely broken. However, there exist cryptosystems from this family that are widely believed to be secure.
The homomorphic properties of their scheme are used to build a kind of multi-user searchable encryption in their main product, Kodex. A user allows other users to search on their documents by homomorphically sharing document search keys. I don't know all the details, but my guess is this specific 'share' operation can be done efficiently because they avoid evaluating a circuit and just do stuff directly.
As with all shiny new things in cryptography, caution is called for. Editorializing a bit, I see little reason to believe this is snake oil. [Indeed, if they were only looking for a rubber stamp from Faugere and Perret it seems like something would have been made public already.] However, perfectly honest and forthright people have built many, many broken cryptosystems, so I recommend taking any claims with a moderately-sized grain of salt until given reason to do otherwise.