My question is: Why did they think they would win?
It's 3 countries with decent-sized (but not huge) populations vs. the whole world. But on the other side was the rest of the world, which easily more powerful. Anyone could have seen that they were severely outnumbered and overpowered, and that they were inevitably going to lose. It's obvious; it was a world War with 3 countries on one side, and the rest of the world on the other side. No doubt those 3 countries were powerful, but on the other side they had the US, the British empire, and the USSR which combined were more powerful easily. They also had a big population advantage with china and india and all the other countries. Even if the Axis managed to temporarily win, and occupy the other countries, it wouldn't last long because they eventually would be overthrown.
So why did they fight in what seems like a losing battle?
I can give a brief answer regarding Japan in two parts. The first is why Japan went to war in the first place, and the second is what Japan’s aims actually were. Although it’s commonly known that Japan screwed up by getting into a war with the USA, what is less mentioned is that Japan was not actually planning to conquer the USA.
In early 1940, Japan was stuck in a war with China it could not seem to finish. While it won a lot of battles, China was simply too big for it to completely conquer, and it stubbornly refused to surrender. To effectively pursue the war, Japan needed raw materials like oil, and it was extremely dependent on supplies controlled by the Western powers.
To Japan’s south, Southeast Asia had large supplies of raw materials, especially oil in the Dutch colonies (present day Indonesia).
When France fell in June 1940, it seemed to open a window of opportunity for the Japanese Empire. France and the Netherlands had been conquered and Britain seemed certain to fall. Practically the whole of Southeast Asia was left without the protection of the Western powers, which meant it would be easy to march in and grab the whole place.
The Japanese began by occupying French Indochina. However, this was a step too far for the Americans, who, along with Australia, Britain and the Dutch government in exile, introduced a trade embargo against Japan. This cut off most of Japan’s oil supply, so now Japan had a choice. It could either give in to American demands and withdraw from Indochina and China, or it could double down and go to war, and we all know what Japan decided.
However, less known is Japan’s aim in war. It was not actually planning to conquer the whole of Asia, let alone the world. What they hoped for was to establish a defended perimeter ranging from the northern Pacific to the Burma-India border. Ground and air forces would hold defensive strongpoints along this perimeter. They would be supported by the mobile striking power of the Imperial Japanese Navy which could move rapidly across maritime Southeast Asia to wherever it was needed. And that was it. The Japanese weren’t planning to go further than Southeast Asia - they weren’t even planning to conquer India or Australia. They assumed that the weakness of the Allies would allow them to conquer fast and then switch to the defensive. Eventually the Allies, especially the Americans, would get tired of trying to break through this perimeter and agree to peace talks, during which Japan would be allowed to retain its newly acquired colonies. Meanwhile, raw materials would flow within this perimeter from Southeast Asia to be used by Japanese forces to finish the war in China.
Of course, this plan didn’t work out. It relied on the Allies giving up and suing for peace, but the Japanese were unable to land the kind of knockout blow on either the Americans or the British that, say, Napoleon used to be able to land to force his foes to the negotiating table. But, the point is, Japan didn’t enter the war expecting to conquer the world. It simply expected, as crazy as it sounds in hindsight, to establish an enormous defensive perimeter it could sit behind and force its foes to give up.