Historically, Bessarabia had closer connections to Romania (the Bessarrabians are essentially a Romanian people) and wasn't really an area within the Russian sphere of influence. The Dneister River was the eastern border of Bessarabia, which was part of Romania prior to WW2.
The Molotov-Ribbentrop Pact with the Nazi's stated that the Soviet's would get Bessarabia, which became the Moldovan SSR within the Soviet Union. Even though Bessarabia had never extended across the Dneister, the Transneister area eventually got folded into the Moldovan SSR within the Soviet Union. In the late 1980's, as Romanian and Moldovan nationalists came to power and began stripping ethnic Russians of recognition and rights, this became a real problem in Transneistria which had never been Romanian and had been ethnically Russian and Ukrainian for centuries.
In 1990, while the entire region was still part of the Soviet Union, Transnistria declared its independence from the Moldovan Soviet Socialist Republic and became the Transnistrian SSR. Facing the very real possibility that this could lead to a civil war within the Soviet Union, Gorbechev issued an order invalidating the Transnistrian SSR and declaring that it was part of the Moldovan SSR. The secessionists never disarmed, however, and the Moldovan SSR never controlled the territory again.
In 1992, Moldova declared its independence from the Soviet Union. In that declaration, they declared that the Molotov-Ribbentrop Pact was illegal and that the Moldovan SSR was illegal and illegitimate. It disavowed any connections with the USSR and charted an independent course for the country (there are still large unionist movements in both Romania and Moldova pushing to rejoin the countries today). Transnistria, which had been de facto independent for a couple of years at that point, seized on that statement . They pointed out that Transnistria was only joined to Moldova due to the Molotov-Ribbentrop Pact, and that by disavowing the results and borders of that pact, they were also disavowing their claim to Transnistria. Because Transnistria had not historically been part of Bessarabia and had been essentially independent since 1990 anyway, they just declared their independence outright.
Predictably, this led to war. Eventually, both sides signed a ceasefire agreement that asked Russia to establish a peacekeeping force in the area to keep fighting from breaking out again. The original agreement said that the Russian forces would stay in place until both sides agreed that it was OK to remove them. To date, the ethnic Russians in Transnistria have not given that consent.
Russia and Moldova signed a treaty in 1994 stating that the region would be ceded to Moldova and that the Russian troops would be withdrawn, but that treaty was never ratified and never entered into force. The Istanbul Accords also would have committed Russia to pulling out its troops, but those were also never ratified. Russia's ratification of the CFE treaty obligated them to remove their heavy weapons from that region, which they did comply with.
It's Russia's position today that, barring any ratified treaties saying otherwise, that the original ceasefire terms Moldova agreed to in 1992 still apply. Under that treaty, they say that they are not obligated to leave until the Transnistrian rebel government asks them to do so.
And unlike the situations in Georgia and Ukraine, the current nation of Moldova has never actually controlled Transnistria. It's been essentially independent since both territories were still part of the Soviet Union.
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