Extremely quick response here:
The term LLJ is thrown about quite a bit to describe separate things. A strict definition of a Low-Level Jet is a river of fast moving air in lower levels of the atmosphere in which the magnitude of the wind decreases with height. A "true" low-level jet can form (and persist) during the daylight hours in and around intense low-pressure systems.
What is probably most commonly referred to as a LLJ on this forum (and I suspect in the SPC products) would be more accurately described as a "nocturnal boundary layer wind maximum". (One theory states that) This occurs after sunset when the thermal wind vector reverses direction (owing to a change in the differential heating on a sloping terrain) and now acts in the same direction as the true wind.
In simpler terms, the thermal wind (actually not a wind, but a shear vector) is oriented such that on a horizontal surface the warmer air is on the right and colder air is on the left.
Using OK as an example, the terrain in the TX panhandle is higher than the terrain in eastern OK. Assume a constant surface temperature across the region and a uniform south wind. If you move up to a certain low-level height above sea-level (so that you are on a constant horizontal surface with respect to sea-level) you would expect the temperature during the day to be warmer at this height in the higher terrain than in the lower terrain owing to the fact you are closer to the warmer surface in the higher terrain. In this event, the thermal wind would be oriented from north to south...while the true wind is oriented south to north. Thus, during the day the two "winds" oppose each other.
Now, at sunset, the ground cools faster than the air just above the ground which cools faster than the air just above that (and so on). So using the same example, you would expect the temperature at our constant height to be cooler after dark in the higher terrain than in the lower terrain. This means the thermal wind has reversed direction and is now acting in the same direction as the true wind. It is when the thermal wind shear vector and the true wind vector align that the "LLJ" (really the nocturnal boundary layer wind maximum) kicks in.
I'm sure I misspoke somewhere in there so someone smarter than I might want to chime in as well.