It's my understanding that during, for example, the Napoleonic Wars, armies were singular groups of soldiers who would march around in the countryside, and battle would commence when two opposing groups would encounter each other.

To an extent you're correct, but by the time Napoleon came to power, when the French Army and the rest of Europe were beginning to institute conscription and raise large, mass armies, warfare had already begun to change. Battles like Wagram in 1809, Borodino in 1812 and Leipzig in 1813 (the largest battle in history until WWI) involved hundreds of thousands of men between either side and the battle lines could stretch for miles. Wagram and Borodino could be seen as particularly prescient, as the flat battlefield at Wagram allowed French and Austrian artillery to wreak havoc, while the French suffered heavy losses at Borodino trying to drive the Russians out of their field fortifications at the Grand Redoubt.

How did it then become that by World War 2 there were wide, thousand plus mile fronts with millions of men fighting essentially non-stop?

By 1914, Europe and the Western World, and much of the rest of the world by proxy, had gone through the Second Industrial Revolution, which saw huge strides in heavy industry, chemical industries, medicine and science. The result was that the population of Europe in particular grew considerably, while Armies kept pace by instituting conscription as well as maintaining large standing armies in the hundreds of thousands, while establishing General Staffs along the lines of Prussia (Germany). Combined with railroads, which allowed large armies and their supplies to be moved quickly over long distances, which now allowed for campaigns to be fought over much greater distances, with much larger and better armed armies, which could in turn be sustained in the field by the well-developed industries of their states.

Secondarily, how did this impact military doctrine?

After WWI it spurred the development of the Operational level of War, in between the Tactical and Strategic levels. This was done first in the Soviet Union by Tukhachevsky and his colleagues, and spread to the rest of Europe and North America by WWII. In the 19th century, it forced changes in tactics, with all armies emphasizing dispersal on the battlefield for infantry, dispersal and indirect fire for artillery (which by WWI had longer ranges and could fire rapidly), and retraining in mounted and dismounted warfare for cavalry, as well as the addition of horse artillery and machine guns. If you want a good book which covers these changes in the German context, Antulio J. Ecchevaria's book After Clausewitz is excellent.