I am going to focus on Japan's path to war, as what happened in Nanking is something that I am not well read enough on to really offer meaningful commentary.

With that said, while we in the West tend to date "World War II" as beginning with the German invasion of Poland on 1 September, 1939, Japan had been at war since July 1937 in the most limited sense and since September 1931 if you stretch to include the Japanese invasion of Manchuria. This war was waged almost entirely independently of what Germany was doing in Europe. Notably, when the Germans invaded the Soviet Union in the summer of 1941, the Japanese did not join in, rather seeking to continue their plans for operations southward against the European colonial possession of southeast Asia, maintaining their non-agression pact with the Soviet Union. Asking why Japan participated in World War II is perhaps a bit misleading, as it tends to ignore Japan's wars earlier in the 1930s. You can probably make a good argument that it was Japan that started World War II, rather than Germany, but that's neither here nor there. Instead, let's focus on why Japan went to war at all.

If I were to summarise why Japan went to war in a single sentence, I would say that it was dragged along by its own military. Most discussions of Japan in World War II will focus on resources--especially oil--and while the desire for an autarkic Japan, independent of foreign imports to sustain itself, was a major rallying call within the Imperial Japanese Army (IJA) in the interwar period, the officers who were most in favor of expanding Japan's access to resources were also against war. Or at least war when it broke out. The need for resources is absolutely important, but perhaps more importantly as an immediate cause for action was the tendency of IJA officers in the field to "take matters into their own hands" and take actions that drew Japan into a war despite being counter to policy--or even direct orders--from their own superiors and other policy makers in Tokyo.

To understand why Japan went to war with China, let us go back to the end of World War I. /u/starwarsnerd222 and I discuss this in some more detail here, but World War I served as a wake up call for leadership in the IJA. The IJA had built itself around the idea of a short, quick, decisive war, as Japan could afford nothing else, yet World War I seemed to indicate that future wars would be long affairs, without easy access to neutral sources of credit or resources. Analysis of the Japanese economy showed that it was desperately short of materials and industrial capacity necessary to sustain a long war. A group of officers within the IJA termed the "total war" officers began pushing for a massive industrialisation campaign that would dramatically improve Japan's economic base for supporting total war. Yet, even under the most favorable estimates, the Japanese empire simply lacked the mineral wealth needed. Nearby Manchuria potentially held that wealth. Interestingly, total war officers--many of whom were part of the General Staff in Tokyo--were very opposed to war...or at least war now. Rather, they wanted to avoid antagonising foreign powers--from whom Japan would need to import machinery and resources for its industrial build up--and direct resources into further industrial expansion. The invasion of Manchuria, at least, seemed to go well. It proved to be a short campaign, and brought Japan access to substantial iron and coal deposits that could fuel Japanese need for steel and even provide the base material for synthetic oil production. Yet, more was necessary, and Japan eyed the additional resources of northern China. The total war officers sought better relations with the KMT government in Nanjing in order to provide Japan concessions in northern China that would enable exploitation of those resources, while simultaneously there was substantial intermingling of Japanese and Chinese troops in the area. This provided the kindling that the Marco Polo Bridge Incident in July 1937 would ignite.

The Marco Polo Bridge Incident began as a simple skirmish between troops over a missing Japanese soldier. Yet, Japanese officers on the ground responded to these skirmishes by escalation, omitting more forces into the fighting and expanding the scale of the conflict, even as the government in Tokyo attempted to reel them back in. Ultimately, as the conflict in northern China began to escalate, the Army and the government decided to try and resolve the issue by force. A swift campaign would secure Japanese access to the resources of northern China without the need to come to terms with rising Chinese nationalism that was driving a harder line from Nanjing. Ultimately, of course, hopes for a short decisive campaign were unfounded, and the war quickly escalated. Japanese forces advanced down railways from the Beijing area towards Wuhan and Nanjing, while a second front opened in Shanghai in August, with more hard fighting that seemed to indicate that Chiang's Nationalist government was not about to simply concede to Japanese demands. The war in China completely derailed plans by total war officers for industrial expansion, as the demands of war sucked up the resources that had been earmarked for construction. More worryingly, the war began to seriously sour relationships between Japan and the United States, which saw the Japanese war in China as an effort by Japan to establish an exclusive economic area that would exclude other nations, in contravention to the U.S. Open Door policy.

Ultimately, the entry of Japan into the wider war via its attack on the European colonial powers and the United States has to be understood in the context of the war in China. By the time Germany invaded Poland, it had been ongoing for nearly two years and by the time of Pearl Harbor, Japan had been at war for nearly four years. The War in China had proven intractable. Japanese forces had pushed deep into China and occupied many of its key cities, but the Nationalist government refused to surrender. As the war continued, the Army demanded further escalation: more troops with more extensive operations to finally "win the war". Japan believed that China's recalcitrance was due to external support from foreign powers, and indeed the US had extended Chiang's government substantial credit to sustain his war effort against Japan. Japan thus sought to end the war in China by isolating China, which involved moving further south into French Indochina after the Fall of France. The move into Indochina led to even more stringent US economic measures against Japan, with the oil embargo and asset freeze that seemed to pose an existential threat to Japan. In order to come to a diplomatic settlement, the US demanded Japanese withdrawal from China. For Japan, to withdraw under US pressure without substantial concessions from the Chinese was politically impossible. The Army in particular would simply not agree to such a withdrawal, and if the government were to try and force it, it would risk a coup.

To add onto this, Japan saw the balance of power in the Pacific slowly but irrevocably tilting against it. The US naval arms build up under the Two Ocean Navy Act threatened to overwhelm Japan with the sheer scale of American naval construction. With the US oil embargo and asset freeze in place, with every day that passed, the US grew stronger and Japan grew weaker. Within a year, Japan would be incapable of waging a war even if it wished. This ticking clock pushed a sense of "use it or lose it" amongst Japanese leadership, and the temptation of moving south against European colonies denuded of their defences due to German successes in Europe was incredibly tempting. By moving south, Japan could not only secure the resources it would need to sustain the war, but it would also serve to further isolate China from external aid, which should help bring the war there to a conclusion. When compared with the alternatives of an Army coup should the Japanese give in to US demands or a slow withering if they tried to endure the US sanctions, Japanese leadership opted for a further escalation of the war to include the European powers and the United States.

Thus, if I were to summarise why Japan expanded its war from China, to include the United States and the European colonial powers, I would say that it was predominantly due to desperation on Japan's part seeking out the only possible path--no matter how narrow--from an impossible situation. Granted, the impossible situation was one of its own crafting, but the only way out of it was to risk everything on forcing the United States to come to a settlement with Japan over the future of East Asia. It was a risky gamble and every decision maker in Tokyo knew it, but with the situation they had found themselves in, war with the United States was the only option that would be acceptable to all power brokers.

I hope this has helped to answer your question. Please feel free to ask any follow ups.