It largely depended on the country in question, although the process by which they joined the Axis was similar in many cases. Romania, for example, was mostly an ally of convenience. Originally Romania tried for a position of neutrality, but after a coup and the Soviet annexation of Bessarabia, the country sided with the Axis. Under Ion Antonescu, the country played a pretty significant role on the Eastern Front. In fact, while the German warmachine was the principal power during the war, Romanian involvement and contribution was quite enormous, with their armies playing a key role during Operation Barbarossa. Towards the end of the war though, with support for the war dwindling Antonescu was ousted in a coup and Romania switched sides to the Allies.

Hungary, much like Romania, was originally reluctant to enter the war on the side of the Axis. That said, with the German annexations of Czechoslovakia and Western Poland, and some pressure from Hitler, Hungary eventually joined the Axis, just in time for the invasions of Yugoslavia and the Soviet Union. Overall, Hungarian involvement was significant, with Hungarian forces being present at many of the major battles on the Eastern Front. That being said, when the tides of war turned, and the Soviets began retaking massive amounts of lost ground, the Hungarian government attempted to negotiate a peace treaty with the Allies. Germany uncovered this, and subsequently occupied the country, albeit not for long due to the advancing Soviets.

As for Bulgaria, it's pretty much the same story as Hungary and Romania. An initial phase of neutrality, followed by joining the Axis and switching sides before the end of the war. That being said, Bulgaria was a bit more of an unwilling ally than the others. In Bulgaria's case, the main reason they joined was ironically due to Italy's military failures in Greece. Hitler originally demanded Bulgaria join the Axis so that he could move German battalions into Greece via Bulgaria to salvage the situation there. The Bulgarian government conceded, albeit somewhat reluctantly. There was a fair amount of dissent amongst Bulgarians throughout the war, with significant efforts being made to save its Jewish population and stronger resistance movements than was typical in other Axis-aligned countries. Towards the end of the war, in 1944, a Soviet-supported coup took down the Bulgarian government and allowed the country to join the Allies.

So yeah, the circumstances varied by country, but ultimately came down to them being originally neutral, then joining the Axis, and finally either switching sides or being invaded.