What would cause a polar vortex to become unstable a split into 3 smaller ones? How rare is this and what long term effects could this cause if the vortex stays unstable?
There are two key intermittent mid latitude circulation patterns during boreal winter.
The other one is sudden stratospheric warming, where the winds cause a disturbance such that the temperature in the stratosphere (the layer above the typical weather layer, the troposphere) warms significantly. To view this one needs to look at the temperature change over the poles, typically by looking at the 10 hPa isobaric surface. If the temperature change is around 50 to 70 degrees Kelvin over an area of synoptic scale and the temperature change is rapid, over a period of 3-4 days, then you have the possibility the weather can be impacted by a sudden stratospheric warming event.
The possibility is that a sudden stratospheric warming in January 2019 could cause the stratospheric polar vortex to split. Based on the peer reviewed reference The Influence of Stratospheric Vortex Displacements and Splits on Surface Climate it is actually the troposphere that causes polar stratospheric variability through vertically propagating Rossby waves. The math behind that is quite complex but it also feeds off linear theory (which can be seen in the quasi-geostrophic Potential Vorticity equation). Some of these vertically propagating waves can travel all the way up into the mesosphere. These lecture notes on Rossby waves mention that Rossby waves cannot propagate vertically if the mean zonal winds are easterly or if they are westerly and exceed a certain speed. It is only the wintertime atmosphere that permits Rossby waves to reach the stratosphere and deposit energy and hence sudden stratospheric warmings are only observed after boreal fall.
In the northern hemisphere most vertically propagating Rossby waves arise due to the topography. In the southern hemisphere the sudden stratospheric warmings are a rarity due to lack of such topographical features. The last sudden stratospheric warming over the southern hemisphere was in 2002.
The resulting stratospheric anomalies can in return influence surface climate but the mechanism on how exactly this occurs is the still the matter of scientific research. One of the theories that is currently popular is the wave mean flow interactions. This can be easily summarized as concerning how planetary scale Rossby waves influence the circulating zonal flows around a planet. The other main theory is wave reflections at the tropopause.
So what is the net result of either of the above two possibilities? Either way they affect mid-latitude storms. Storms could either become more intense, shift equatorward, or there could be extremely cold air advection spells as the stratospheric air gets mixed into the troposheric air.
As to the rarity of polar vortex splits - they are coupled to those sudden stratospheric warmings. If you look towards the climatology of SSWs, observations show that they occur more during El Niños and La Niñas than during neutral conditions. Sudden stratospheric warmings happen almost every other year in the NH. However sometimes the polar vortex isn't split during these events but is merely pushed equatorward.
The long term effect a polar vortex split could be extended spells of extremely cold weather in the affected areas, and when the vortex reforms on into summer it would be shrunk in size from the dispersed energy giving rise to greater easterly wind circulation near the poles.
Now that the reanalysis data is available from January 2019 I want to add some visuals for the Dec-2018/Jan 2019 polar vortex split and ongoing cold wave.
The OP's link, plus this SlashGear article on the polar vortex split provide GEFS and GFS forecast data, pointing to a three-way split of the polar vortex. I went through reanalysis data from the period of Dec 18th 2018 through Jan 30th 2019 and there appears to be two separate candidate dates for a three-way split of the stratospheric polar vortex; it really depends on whether one considers the small vortex over the North Pacific as one of the split vortices or not on Jan 5th 2019. The 5th January event is the one that most closely resembles that GFS forecast data. In saying this, the reanalysis data is averaged over a full day as opposed to the single point in time of the GFS data. In the geopotential height anomaly there is a clear three way split but one must remember that the anomalies are calculated vis-a-vis the climatological period 1981-2010. Either way this will require more detailed research before the three-way split is confirmed or not.
Finally from the reanalysis data it appears that, in contrast to the 2018 event, this year's event points to significant instability in the dynamics of the polar vortex. There are clear suggestions that the polar vortex is breaking and reforming again over short time scales, which is interesting.