This answer relates to a tassel of sufficient length to wrap the wielder's wrist. I typically use one with a string length 9 inches, but prefer 12", which are harder to find, but, perhaps counter-intuitively, allow better control. Tassels with no string attached directly to the pommel have no function beyond aesthetics.
This is typically the generic given answer, the idea that the moving tassel can draw the eye and distract a competitor. However, as serious practitioners must train their gaze to be cognizant of all details of the opponent and surroundings, and is mostly concerned with the opponent's blade, stance, and orientation, this distraction function is only likely to be a factor against a novice.
This is another common explanation, and has some utility. There are certain counters that allow for the tassel to directed at an opponent's eyes, which, if it makes contact, will result in some stinging and tearing up. "Sparrow carries the soil" from Tai Chi sword is another technique can be executed such that the tassel leads into the face of an opponent. As these techniques can only be utilized at extremely close range, it can be advantageous if executed successfully since the opponent is well within range of a finishing cut.
- Provide counter-balance for certain strikes
For some techniques, putting energy into the tassel can provide a counterbalance for a cut, "spreading wings" as an example, which perhaps provides some small advantage. This is because, although the tassel is typically light, sufficient inertia can be generated to balance against a light, flexible blade.
Loose fabric is well known for its ability to snag the point of blades. Tassels can be used in a similar manner, as even edges don't do well against loose fabric. Where a tassel is controlled properly such that the string is rarely entirely taut, it can redirect an opponent's blade out of the intended attack vector. (If entirely taut, the tassel string can be cut.)
- Wrapping a pole-based weapon to set up a cut
This is something I was not taught, but validated in research. The tassel can wrap a weapon like a spear, and the fencer, having superior leverage, can simultaneously redirect the pole weapon while making a decisive cut. (In this case, the tassel must be attached around bottom of the handle, above the pommel, not the weak fastener at the end of modern screw-based tangs, or it will be ripped off.)
- Wrapping one's own wrist intentionally to provide stronger grip.
This is something you see rarely, likely because it also has the effect of reducing wrist mobility, and wrist flexibility is arguably the primary requirement of Chinese straight sword.
- Using a tassel makes the practice significantly more difficult
This is because the practitioner can no longer focus only on movement of and energy to the blade, but must also precisely control the movement of the pommel, avoiding allowing too much energy into the tassel. Failure to properly control the tassel can result in wrapping the wrist (although even high-level masters can have this happen on occasion. In this instance, the practitioner takes it as an opportunity to naturally unwrap the tassel in subsequent movements, without compromising martial technique.)
- Demonstrates level of control
In the same way looking at a practitioner's off-hand in weapons forms is a simple way to gauge their skill, observing a the tassel reveals the level of control. Nearly all of the energy should be in the blade—if the tassel has too much movement, it indicates energy bleed to the wrong place.
Even in energetic cuts, one avoids the tassel spinning like a propeller, and seeks to direct that energy in a line, such that the tassel bounces once rather than spins. At a very high level, a master can execute a cut with great focus (fa jin 發勁) where the tassel only expresses a minimal bounce, with no linear extension of the string at all.
- Gauge one's own level of calm in internal styles
It's not uncommon for the mind not to be clear during practice, which can result in insufficient calm for Tai Chi and Wudang sword. Seeing one's tassel bouncing around in a poorly controlled manner is a clear signal to the practitioner.
(Note that calm doesn't necessarily mean slow movement at a steady pace throughout—although internal sword is often practiced this way. Real fencing involves tempo, such that an advanced Wudang practitioner will vary pacing in both practice and exhibition—some movements slow, some quick and lively.)
Long tassel sword is a special case. In contemporary Wushu, this refers to a string+tassel fringe longer than the blade, but the traditional form I was taught utilizes a string that is itself significantly longer than the blade, typically more than twice as long, and this informs my understanding of tassel in general.