Why a smooth stream of water breaks up as it falls
Surface tension against acceleration, and why the break happens where it does.
A falling stream of water breaks up into separate drops because a long, thin column of liquid is an unstable shape that surface tension is constantly trying to correct, and any small unevenness in the column's width, always present however smooth it looks, grows steadily as the water falls until it pinches the stream apart into a string of drops.
A tap left running just enough to keep a glassy, unbroken column falling from the spout always breaks into chaos at roughly the same distance below it, never right at the tap and never an arbitrary distance down, always somewhere in between that looks, at a glance, oddly fixed.
Why a thin column of liquid cannot stay a column
Surface tension behaves like a thin, stretched skin sitting on the outside of any body of liquid, constantly trying to pull that liquid into whatever shape has the least possible surface area for its volume, which for a free-floating blob is a sphere. A long, unbroken column of falling water is about as far from that preferred shape as a given amount of liquid can be arranged, since a thin cylinder has far more surface area for its volume than a string of separate spheres holding the same total amount of water would have.
The column survives as long as it does only because nothing has yet given the surface tension an opening to act on. No falling stream is ever perfectly uniform in width, though. Tiny random unevenness is always present, a slightly narrower pinch here, a slightly fatter bulge there, too small to see with the naked eye at first.
Surface tension pulls harder on the narrower sections, since a thinner neck of liquid has more curvature for its size and surface tension responds more strongly to sharper curves. That stronger pull squeezes the narrow sections even narrower, while the water squeezed out of them bulges the neighbouring wider sections even further. The unevenness that started too small to notice grows larger the further the stream falls, until the narrow sections pinch closed completely and the column separates into a string of individual drops, each one settling toward the rounder shape the whole column was always being pulled toward.
Nothing external needs to disturb the stream for this to happen. No draught and no vibration in the tap is required, because the instability is built into the shape of a thin cylinder of liquid itself, and even a perfectly smooth column would find some infinitesimally small unevenness to amplify.
Beads of rain on a washing line
Rain or dew settling onto a horizontal length of string, wire, or a washing line rarely coats it as a smooth, even film. It gathers instead into a row of separate beads spaced out along the line, each one plump and rounded.
A thin, even film of water clinging all the way around a narrow line is exactly as unstable a shape as a falling column of water, for the same reason: too much surface area for too little volume. Surface tension pulls that film apart into separate beads by the same route it pulls a falling stream apart into drops, favouring whichever narrow, high-curvature necks exist to squeeze themselves shut while the water gathers into rounder pockets on either side.
The washing line offers a version of the process a person can inspect while it sits still. The falling stream shows the identical instability happening too fast to follow by eye, the smooth column above the break and the scattered drops below it being two snapshots of the same process caught at different moments.
Why the break always lands in the same place
Not every ripple along the column grows at the same rate. A bulge-and-neck pattern shorter than the column's own circumference cannot grow at all, because squeezing liquid into bulges that close together would add surface rather than remove it. Longer patterns do grow, and the one that grows fastest has a wavelength of roughly four and a half times the stream's diameter. That fastest-growing pattern wins the race and sets both where the drops form and how big they are, which is why the drops falling from a steady stream come out so evenly sized and evenly spaced.
The break point looks oddly fixed for the same reason. For a given flow rate and nozzle width, the tiny initial unevenness takes a fairly consistent amount of falling time to grow large enough to pinch the column shut, and a stream moving at a steady speed covers the same distance in that time on every occasion the tap is watched.
Honey, syrup, and fast jets
This account of a plain water stream does not carry over unchanged to a much more viscous liquid like honey or golden syrup poured from a spoon. A thicker liquid resists the internal flow needed to move itself out of the narrow, pinching sections and into the bulging ones, which is why a thin stream of honey can stretch out into a long, wavering thread far longer than a stream of water ever manages before it finally breaks. The same surface-tension instability is still at work underneath, only fighting a liquid that is far more reluctant to rearrange itself quickly enough to obey it.
A fast-moving jet, such as water forced hard through a narrow nozzle, breaks up by a messier route too. Once the water is moving quickly enough, the surrounding air starts dragging and buffeting its surface, tearing the stream apart through turbulence well before the slower, tidier pinching process described above has time to run its course.