Why do cats blep?
The word “blep” refers to a cat sticking her tongue out slightly. The face is passive. The body is also fairly passive and still. There appears to be nothing particular going on except that the cat is probably thinking. Humans do the same thing (see photo below).

Top photo in public domain. Bottom photo by Michael
You won’t find reasons for this slightly strange feline behavior in the books. Therefore we have to work it out for ourselves. There are various theories.
Flehmen Response
One person thinks that the cat blep signifies that she is tasting the air and passing the scent molecules to the Jacobson’s organ (aka vomeronasal organ) in the roof of the mouth which is a highly sensitive olfactory organ making the cat able to smell things very accurately.
I do not think that this is the answer. I don’t think the cat blep has anything to do with the Flehmen Response as it is called.
I believe this because when a cat leaves her tongue slightly out of the mouth, as mentioned, the cat is passive. When a cat indulges in the Flehmen Response the mouth is slightly open (or more for the snow leopard) forming a sort of grimace and the cat has a purposeful and intense appearance. The cat is working and the cat’s brain is processing the odour. This leads to a completely facial appearance than that which see with the cat blep.
Smile?
Another person on Reddit.com thinks that this is a form of cat smile. I don’t believe that either.
No significance other than the cat is relaxed and thinking
So why do cats blep? The cat with whom I live at the moment doesn’t blep. A female cat with whom I lived some years ago did it occasionally. On one occasion I gently tapped the tip of her tongue and it popped back into her mouth. If was if I had pushed it back. There was no other reaction. As I said, when a cat does the blep she/he is passive.
I think that the cat is doing nothing when performing the cat blep except think. The cat is relaxed and it may be that this passivity and relaxation which results in the tongue falling out of the mouth. It is not a distraction behavior. The cat is not tasting anything and in my opinion it does not signify anything except perhaps, as mentioned, the cat is thinking. It is the posture of a relaxed cat. It is what it is. Not every feline behavior has to have a specific purpose which has evolved over many years.
Human Thinking with Tongue Out
It is not uncommon for humans to do exactly the same thing when thinking:

Afghan girl thinking. Photo in public domain