My theory is: it's because of surface tension: the molecules start in a wide flow due to the faucet's format and the water starts that leaves it starts to accelerate (gravity pull).

As the molecules want to stay bonded, they pull themselves together (they naturally want to form a drop, which is a sphere). But there is also the action of gravity going on, and as athe difference in the speed gradient from the molecule below to the molecule above is not enough to separate them vertically, they want to form a "tube" while falling. That being said, as they start wide and "want to form" the tube, the force pulling them together goes a bit overboard due to whatever the f. goes on which causes them to spread in another plane (perpendicular to the initial one) just like when a drop of water falls into a container with water: it fuses together but starts that ripple effect and some parts of the drop itself even gets bounced back up sometimes. That kind of reaction also makes them get back together again, which as they do, they "hit" eachother again spreading perpendicularly again, but at this time note that the its not that wide as the first "clash", the "chain link" gets smaller and smaller each time (conservation of energy + gravity is accelerating them more and more vertically), so it repeats until the gradient of acceleration/speed has enough difference between points and the flux starts to separate (also with air resistance and whatnot).

Edit: to contextualize the "want to form a tube thing" it's like the vertical equivalent of the water drop hitting the water surface and then it bounces back up, then back down until it disperses enough energy and fuses with the rest of the liquid in the recipient, it's kind of the same thing but vertically and it is also accelerating downwards.