If I had to guess, it's sediment that gets re-suspended when handling and moving the bottles.
When storing (aging) your bottled ciders (or other fermented, then bottled, beverages), usually under cool conditions, they'll get clearer over time due to residual yeast and other suspended solids slowly precipitating (dropping out). They're consumable and not dangerous to your health. It's more of an esthetic value to have (crystal) clear beverages.*
Be very gentle when handling, opening, and dispensing from them, to prevent that remixing from happening.
When pouring out, do it slowly in one steady stream, without tilting back or "
glugging." That way the sediment will not mix and stays low in the bottle.
Toward the end of your slow and controlled pour you may see a narrow stream of that sediment slowly moving toward the neck. Before it reaches the opening, right before it would pour into the glass, tilt the bottle back, quickly, in one smooth motion.
The result: clear cider in your glass! And a little bit (~1/4") of cloudy cider left behind on the bottom of the bottle.
The cloudy cider that's left in the bottle is also drinkable, may have some extra flavor, which can be an acquired taste.*
* Alas, not everyone's digestive system reacts positively to drinking the yeast/sediment or even "cloudy" beverages.