Italy was a fascist state and thus a natural political ally of Nazi Germany located in the strategically important Mediterranean basin. With the Empire of Japan Nazi Germany had common enemies.
The alliance of Germany and Japan had surprisingly little practical significance. Just by looking at the map one could've thought Japan and Germany could've attacked the Soviet Union in concert and defeated it, putting both and Germany in particular in a much better strategic position than in real history. But Japan and the Soviet Union had already clashed in a brief war consisting of battles in Khalkhin Gol, Mongolia in the summer of 1938. The result was a complete rout for the Japanese. Since then, the Japanese left the Soviet Union alone and were happy to purchase oil and raw materials from the Soviet Union. What the two powers had was effectively a truce the Soviet Union broke only in early August 1945 right before the war ended in Japan's surrender after the nuclear attacks on its soil. In that little known campaign, the Soviet Union crushed the Japanese Kwantung Army occupying Manchuria in a massive operation advancing all the way into the northern part of Korea.
Japan had another reason to be reluctant to help Germany by attacking the Soviets in the Far East. Japan's war was all about securing access to vital raw materials in Asia, oil in particular, which was found in Dutch-controlled Indonesia, and also rubber. Japan's aim after the United States declared an oil embargo on Japan in August 1941, was to take control of the regions containing strategic raw materials in a swift campaign, which they did, and the build a strong defensive perimeter in the islands, extending far into the southeast to isolate Australia and New Zealand from their allies in North America. The Japanese leadership were hoping to deal the U.S. Navy a heavy blow in the beginning and exact such heavy a toll on the Americans on their way towards the Japanese homeland that they would consider a negotiated peace while allowing Japan to keep access to strategic resources. The purpose of the oil embargo which the Roosevelt administration knew would force the Japanese to take radical action was to get the reluctant American public on board to fight a war to contain the rising might of Japan. The Americans had watched the expansion of Japan's control over China with great concern.
The Japanese effort to defend their vast conquests wasn't sufficient. The main reason Japan lost is the fact that American industrial capacity was second to none in the world. As the war progressed, Japanese war material production was outpaced by America by an order of magnitude and the Americans were capable of fielding well-trained aviators which Japan ran out of in the first years of the war. Also, technologically the Japanese fell behind in the course of the war. The Americans had been overrun in the Philippines in 1942 but the reconquest of the Philippines in 1944 was preceded by a naval and aerial campaign described as the Great Mariana's Turkey Shoot.
In short, the Empire of Japan did not effectively help Germany in the east because their vital interests were elsewhere and their main enemy far too powerful for them to divert any resources against the Soviet Union, a powerful enemy in its own right capable of using tanks to a great effect in the steppes of Mongolia.
It's a great subject, Thanks for your question ..
The Italian alliance has enough reasons. On the Japanese one I'll leave it for the end.
The first thing is that we must not forget that Mussolini came to power long before Hitler. During that time, Hitler saw, learned and copied many of the characteristics of the fascist movement in Italy.
Then, once Hitler came to power, Italy was one of the natural allies, since it was close (after the annexation of Austria they shared border), they had no territorial conflicts (really if there were any in the East of Italy, obtained from Austria-Hungary after the First World War, but Hitler was not interested in the least), had similar political systems (fascist dictatorships) and their areas of interest were close, but not overlapping (Germany wanted Poland and European Russia, while Italy I wanted the Balkans and Africa).
In addition, between Hitler and Mussolini was created a friendship (they were not the "best friends", but if there was some friendship.) We must also bear in mind that the Allies did not fulfill the promises given to the Italians to betray the Triple Alliance in the First World War (they were granted only a few Islands of the Ottoman Empire and the area of Veneto and Istria [Treaty of London (1915)].
On the other hand, the "alliance" (to call it somehow) with the Japanese, was based only on the phrase "The enemy of my enemy is my friend". Both Japan and Germany feared the USSR, and in 1936 they signed the Antikomintern Pact (Italy also joined, among others).
And as for the tripartite pact, Japan adhered because it was a way to get French Indochina without fighting.
But there was nothing else. Germany and Japan looked for their own interests, instead of looking for common interests. No strategic objectives were ever coordinated or there was no military coordination (except to offer some port for submarines).
After all, if Japan had invaded the USSR in 1941 or 1942, the result might have been another [If Stalin left the troops in Siberia, he would not have been able to carry out the 1941 winter counterattack, and in that case, It would have been necessary for Japan to attack, only the "threat" or "possibility" of it, and even if it had taken the troops, because the Japanese advance, although of doubtful quality, would have provoked a war on 2 fronts to the USSR, inevitably absorbing resources that were used against Germany in 1942. It may still have won the USSR, but ... who knows ...
Shared values of totalitarian control of the government and politics of expansion.
Unlike the alliance of USA and Soviet Union, it was born from mutual respect and admiration.
The "Center point powers" formally took the name after the Tripartite Pact was set apart by Germany, Italy, and Japan on 27 September 1940, in Berlin. The settlement was along these lines joined by Hungary (20 November 1940), Romania (23 November 1940), Slovakia (24 November 1940), and Bulgaria.