I have been here before, about a year or year and a half ago when this question was raised. I have only a little add because there have been a number of good, elucidating comments - and some not so elucidating - and some good sources have been cited where valid information can be found so I am not sure what I have to offer will add much, but here it is anyway.
In no way should Low German be considered a dialect. It evolved from one of five or six (depending on how you count them) early Germanic languages, e.g. Gothic (now dead), Old Norse which became the language base for Scandinavia and Iceland, Old Frisian, Old Saxon which is the basis for Low German and very close to Frisian the basis of modern Dutch and Afrikaans, and then two identified as Old High German, the Rhine/Frankish and further south the Aleman.
As has been so well pointed out Germany remained a disunited conglomeration of states and cultural entities until unification was finally achieved in 1872 so these local languages continued to live on until the process of unification was started. And travel was limited so there was little need for the populace to bother with other languages but governments and businesses did have a need for a common method of communications. From this evolved what is often termed "Kanzel" German, or German of the chancelleries/governments. This, along with the language of his home area gave Martin Luther the basis of his "High German" used in his translation of the Bible. This became a major unifying factor in the Protestant parts of Germany but it would take some time. It might be noted that the Bible had already been translated into Low German and in fact, a copy of such a Bible that pre-dates Luther's translation is on display in the museum in the Wartburg where Luther did his translation.
Back to Low German! The oldest extant copy of German law is found in the "Sachsenspiegel" (1220-1235) that was written in Low German. Then from 919 a.d. - 1024 Saxon Kings, later Emperors, ruled Germany and the Holy Roman Empire until succeeded by another German dynasty. Of course their language was that of the Saxons and we can even speculate that had the Ottonian (Saxon) line continued and had succeeded with some sort of German unification Low German would have emerged as the national language. And has been punted out Low German was the language of the Hanseatic League because the main sea ports were of course along the North and Baltic Seas where Low German was spoken.
Even after the moves toward German unification started Low German survived as the daily language of the people in the area of Lower Saxony, and in fact, across Northern Germany. The reason is quite simple; this was sort of a "back water," rural area with little contact by the populace with other areas, much like where I live in an out-of-the-way place on the northern edge of the Ozarks. History passed us by, just as it did the people in the rural areas of Northern Germany. People seldom left their home areas. I have traced my family back to 1628 in Lower Saxony. When one guy moved away from home to chase a woman in the early 1700s but he only moved some 40 miles. >From this point on the family remained within a five mile radius, and most are still there and speaking Low German.
Ours is a community in Missouri settled and built largely by people from the area around Bremervoerde and we still maintain our Low German heritage with an annual Low German Theater. We are frequently visited by groups from Germany and they are amazed to find our Low German speakers speak the language as it was spoken by around 1850 the time of the major immigration where as in Germany the language has grown, as languages do.
I hope this does not add to the confusion.........
Bob Owens (Mostly German but a Welshman crept in somewhere along the line.)