Political expediency, mutual goals, and partially shared ambitions really. For Hitler and Mussolini, at least.

What's interesting is that there was a point where Mussolini wanted to be allied with anybody but Hitler, but the Duce's own political goals and motives would force him into a situation where he had to accept Germany as an alliance. Mussolini knew by the 1930s that Italy was nowhere near ready to fight another world war but he also desired recreating his version of the Roman Empire. Short and dirty this meant war was inevitable. This also means he had to curry alliances, favors, and appeasements with the 'great' European powers (namely Britain, France, and Germany) in order to 'trade up' Italy's reputation. If Italy was seen as a great power, it would give the other nations less course of action and (most importantly) desire to meddle in their affairs.

So Italy would push for pacts and alliances outside the purview of the League of Nations that would explicitly lable Italy as on the same level, in terms of the international system, as the other powers. Most importantly though, these alliances were not to make Italy actually /committed/ to another country (or as foreign minister Dino Grandi viewed it: enslaved). There was an end date in mind to when Italy could stop walking this tight rope: 1935. This was when Germany's Treaty of Versailles obligations were due to expire. Mussolini, Grandi, and the Fascists saw it best to sit out signaling any true intentions until then.

That is of course until Hitler and the Nazis came to power and upset the system Italy had been slowly cultivating. The best laid plans of mice and man, and all.

By 1932 Mussolini had gotten a deal generally worked out between Britain, France, Germany, and Italy. Each state would be allowed to have unrestricted rearmament, with the caveat that each state would be allowed unrestricted inspection. Clauses in the agreement called for increased economic and diplomatic cooperation, 'common thinking' in terms of colonialism, and solidify the four as the de facto hegomonic powers of Europe and Africa. To say this would have been the treaty of the century is an understatement.

When Hiter came into power in 1933 he rejected it out of hand. He did not want the world to know of Germany's rearmament plans, and did not want to be restricted to any arms control. Mussolini took this as a personal slight. Mussolini was also becoming concerned over Hitler's interest in Austria - which was inflamned when the Nazis assassinated Engelbert Dollfuss, the austrofascist dictator of Austria, in 1934. Mussolini immediately deployed the Italian army to the Austrian border and said that if Germany invaded Austria he would invade Germany in kind. Tensions eventually cooled (helped by the German and Austrian Nazi parties severing ties), but the personal damage was done. Mussolini would start to look towards creating alliances AGAINST Germany as he did not view Hitler as having Italy's interests at heart.

Political expediency, and mutual interests would - however - bring the two countries back together.

When Italy invaded Ethiopia in 1935, Germany would be one of the lone nations to support Italy (as Germany had their own Empire building aims, and wanted to continue to undermine the League of Nations). This would lead to a reassessment of Italo-German relations: whatever personal differences existed between the two leaders, the two countries had similar aims. They mainly recognized that they had common enemies towards their ambitions in the Anglo-Franco alliance. While not official, after 1935 Italy and Germany were in a de-facto alliance. In 1936 Mussolini would state that Europe was to rotate on a Berlin-Rome axis. The addition of Italy to the Anti-Comintern Pact in 1937 saw the true alliance between Italy and Germany begin.

In case of Japan it was mostly for military reasons for both parties. While they did have similar racial beliefs, they were polar opposites because they saw their own race as the "master race" respectively.

Japan had been extremely aggressive in their invasions on different countries in East Asia and the pacific since the 19th century. After Commodore Perry forced a trading contract on the previously isolated Japan, they wanted to become a super power in order to be able to stand up against the West and not be forced into any more treaties. As you probably know, during the Meiji Restoration in the latter half of the 19th century and the beginning of the 20th century they occupied multiple isles in the pacific and invaded Korea and Manchuria. They even won against the Soviet Union during the Russo-Japanese war in 1904-1905. Thus, strategically speaking, Japan was a good partner to have for Nazi Germany since Russia had to divide their forces to fight against the Japanese in East Asia and Britain had to dedicate part of their navy to fight against the Japanese in the pacific. Germany also anticipated that the US would eventually join the war and would have to fight in the pacific, which they did.

For Japan, it was simply good to have an ally in the west, so they would have strategic advantages during future invasions.

Interestingly Japan had similar ideas to Nazi Germany's National socialism with "Ultra Nationalism". Both were extremely aggressive in their invasions of neighboring nations. Germany felt they did not have enough room for their people after the treaty of Versailles and Japan lacked resources. The image of the brave, manly and loyal soldier and the woman who is a mother and housewife was extremely prevalent in both ideologies.

As I mentioned before they both saw themselves as the superior race compared to the other. In the case of Japan, they believed that since their emperor was of divine heritage, they were a superior people. On a side note, this sense of unity had only been established during the late 19th/early 20th century. Before, the Japanese found their individual, local communities far more important. Thus, due both sides having extremely nationalistic views they merely saw each other as good allies during war, nothing more, though they had a similar mythos, militaristic structure and sense of nationalism.

There had been cultural exchanges between Germany and Japan for centuries, an example of which is the "German-Japanese Society". In the 1930s they began to make even more cultural exchanges, mostly in the form of people visiting the respective countries. There was even a new Japanese embassy to be built in Berlin. Japan did not have enough attack power to fight against the Red Army at the time, however both nations had clearly anti-communist views and thus Germany believed that there would eventually be a German-Japanese war against the USSR. This strategic partnership eventually led to the Tripartite pact.

Edit: Sources:

Tsurumi Kazuko, Social Change and the Individual: Japan Before and After Defeat in World War II

K.Antoni, P.Pörtner, R.Schneider: Mythos und Ideologie im nationalsozialistischen Deutschland und kaiserlichen Japan der frühen Shôwa Zeit