When the Nazis seized power in 1933, Germany was one of the least motorized societies in Europe. Hitler idolized the motorcar and devoted a considerable drive to increase production, constructing the Autobahns, and propagandizing to the population in order to ingrain motorcar ownership and use as German cultural value. However, due to the contradictory nature of the Nazi "economic recovery" and rearmament campaign between 1933-1939, the motorization of Germany, including the army, was nowhere near complete by the eve of WWII:
As a result of such interventions [in favor of rearmament], the motorway linking Hamburg to Basel was not actually completed until 1962. Moreover, few people had the means to enjoy them before 1939, since Germany was one of the least motorized societies in Europe. In 1935, only 1.6 percent of the population in Germany owned motor vehicles, compared to 4.9 percent in France, 4.5 percent in Britain, and 4.2 percent in Denmark. All these figures were dwarfed by vehicle ownership in the USA, which stood at 20.5 percent, or one in five of the population.
In his [1933] speech at the Berlin motor show, Hitler announced not only the inauguration of the motorway building program but also the promotion of motor sports and the reduction of the tax burden on car ownership. The result was a 40 percent increase in the number of workers in the motor vehicle industry from March to June 1933 alone. Motorcar production doubled from 1932 to 1933 and again by 1935. Well over a quarter of a million cars were now being produced every year, and prices were much lower than they had been at the end of the 1920s.
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Despite the spread of car ownership, however, the motorization of German society had still not got very far by 1939, and to describe it as a powerhouse behind Germany's economic recovery in these years is a considerable exaggeration. By 1938, to be sure, Germany's vehicle production was growing faster than that of any other European country, but there was still only one motor vehicle there per forty-four inhabitants, compared with one for every nineteen in Britain and France. The vast majority of personal travel and the movement of bulk goods was still accounted for by Germany's railway system, Germany's largest employer at the time...
When I speak of the contradictory nature of the Nazi "economic recovery" and rearmament campaign, I speak of how the Nazis focused almost exclusively on rearmament during the period of 1933-1939. It was prioritized above all else, and the industrial demands of meeting the Nazi production targets rapidly exceeded and soon utterly dwarfed available resources. The Nazis pursued a policy of "Autarky", or economic and resource self-sufficiency; but at the same time, the drive to rearmament by 1942, the original goal, to fight a prolonged and potentially two-front war, made Autarky impossible. Such contradictions in both ideological doctrine/rhetoric and in practice were a very common feature of National Socialism.
The production targets set by the Nazi Party grew larger and larger while simultaneously the target date for total rearmament drew nearer and nearer, such as to 1940. The Nazis had to print enormous amounts of currency to run the deficit spending required to import the resources they needed for the rearmament campaign, and this was still not enough. As Hitler had constantly preached about the need for conquering "living space" in the East by invading the Soviet Union, German government finances were pushed to such a disastrous brink that the necessity of launching an aggressive general European War became not only completely inevitable, but always loomed closer and closer, which was the driving force for the Nazis moving the target date for total rearmament closer.
As a result, by the eve of WWII the drive to motorize the Wehrmacht (much less civilian society) was nowhere near complete. The German military still relied extremely heavily on rail travel and horse-drawn vehicles for their logistics. As a result of this in turn, the German military was ironically woefully unprepared to fight a prolonged war of attrition against its enemies, especially the Soviet Union; it was from this logistical and industrial inadequacy that the doctrine of blitzkrieg sprung. The entire grand strategy of Nazi Germany going into its invasions of France and then the Soviet Union utterly relied upon the extremely rapid decapitation and capitulation of those enemies - if the invasions bogged down at all, the German economy would either catastrophically collapsed within a relatively short period of time (in the case of France) or would rapidly become completely outclassed by the enemy's industrial capacity and manpower (in the case of the Soviet Union). While the blitzkrieg against France worked beyond all expectations and allowed Nazi Germany to compensate for its catastrophic fiscal policy through plundering the majority of the European continent, the invasion of the Soviet Union stalled short of its goals, and the tide soon turned against Germany by early 1943 if not earlier.
Whatever propaganda messages about the battle for work might claim, Nazi economic policy was driven by the overwhelming desire on the part of Hitler and the leadership, backed up by the armed forces, to prepare for war. Up to the latter part of 1936, this was conducted in a way that aroused few objections from business; when the Four-Year Plan began to come into effect, however, the drive for rearmament began to outpace the economy's ability to supply it, and business began to chafe under a rapidly tightening net of restrictions and controls...
What Hitler wanted to ensure, however, was that firms competed to fulfill the overall policy aims laid down by himself. Yet those aims were fundamentally contradictory. On the one hand, autarky was designed to prepare Germany for a lengthy war; on the other hand, rearmament was pursued with a headlong abandon that paid scant regard to the dictates of national self-sufficiency. Measured by its own aims, the Nazi regime had only succeeded partially by the summer of 1939. Its preparations for a large-scale war were inadequate, its armaments program incomplete; drastic shortages of raw materials meant that targets for the construction of tanks, ships, planes, and weapons of war were not remotely being met; and the situation was exacerbated by Hitler's program. The answer was plunder...The enormous stresses and strains built up in the German economy between 1933 and 1939 could, Hitler himself explicitly argued on several occasions, ultimately only be resolved by the conquest of living-space in the east.
Source: The Third Reich in Power, by Richard Evans
The German Army was overwhelmingly an infantry-based force. In 1939, the German Army consisted of 85 infantry divisions and three Mountain divisions (infantry with mountaineering training and gear). The typical infantry division of first wave (the best equipped of the initial four "waves" of mobilization) had authorized strengths of 17,734 men, and 4,842 horses. There were only about 1,500 motor vehicles (about a third were just motorcycles) in each division.
Compared to these 88 divisions, the Panzerwaffe consisted of 5 and a partial sixth Panzer divisions, 4 "Light" mechanized divisions (former cavalry divisions), and 4 motorized infantry divisions- about 13 1/2 divisions total. More divisions of all types would be raised during the war, but the ratios wouldn't vary much, even though the plundering of conquered territory included the confiscation of just about anything with a motor (or hooves, for that matter). In fact, by the final two years of the war (late 43 to May 1945), many units were "demotorized" due to wartime losses and petrol shortages.
FYI, you might get better answers if you provide the source of those quotes. History and historiography are inextricable.
Others have answered the part about most of the German army being horse-drawn. Regarding the livestock size, Germany did kill many of them during WW1, but economic censuses indicate that pre-WWI levels (adjusted for territory loss) was achieved during the early 1930s.
http://www.helsinki.fi/iehc2006/papers2/Grant.pdf
Go see figure 9 of this paper about German agriculture. During WW1 the Germans slaughtered officially 5 million pigs (this event is labeled Shcweinemord), yet the pre-war level of pigs was achieved in 1933.