Preservation of individual nerve-fibres look quite extreme, but students of palaeo-sciences are quite familiar with seeing rock-slides containing fossilized tissues, where the contours of every cell is clearly visible.
wikipedia - image, cropped and rotated to save space. It represents a siliceous petrifaction mode of preservation, of a plausibly soft-bodied organism.
The most well-preservation of cells are found in petrifaction or cellular permineralization modes. (Though there are disputes about when to use these 2 terms (such as second one has a broader sense use), the 2 things are same.)
Though your mentioned page did not mentions anything about its plausible mode of preservation, it apparently looks like a compression fossil , which also contain preservation of harder structure such as plant-cuticle, but preservation of individual nerve fibers are really surprising. (However just speculating, may be it is actually a petrifaction fossil or something in-between of petrifaction and compression).
No one of us know, how exactly the fossilization processes take place, but things unanimously accepted are, fossilization requires very very special conditions/ situations, and which is very rare and chance dependent, and slow also.
It has been hypothesized that fossilization requires anoxic environment, source of sedimentary materials (preferably including finest-grain clays), soluble minerals, quiet or no-shake conditions.
For petrifaction-fossils, it has been hypothesized that a source of minerals that would deposit, is mandatory, such as the sulphides, silicates carbonates, sulphates etc of metals like iron, magnesium, calcium, etc. coming through volcanos, hot-springs etc. and also marine sources of salts may have played some role.
In case of silicified petrifaction, presence of soluble silicates are important. An increase in acidity (decrease of pH) can trigger the deposition of insoluble forms, thus may have generated silicified petrifaction.
I've read about one example observed present day, that resembles (and may explain) permineralization-type fossilization, from Stewart- Rothwell's Palaeobotany.
"A park ranger reported finding the remains of a coyote in the outflow of a geyser. He left carcass undisturbed and found, when he returned some four years later, that the remains of coyote bones had already become silicified in the same way as the bones of dinosaurs".
Though this does not include very soft tissue, yet a present-day example of silicification.
All informations based on Paleobotany and the evolution of plants/ 2nd Edition, Wilson N. Stewart and Gar W. Rothwell/ Cambridge University Press.