Russia's interest: it's the ports

It's not their only interest, but understanding the importance of the two ports is one of the keys to understanding Russia's actions:

The Link Between Putin’s Military Campaigns in Syria and Ukraine

[...] Both Sevastopol and Tartus play a role in compensating for Russia’s geographic deficiencies as well. “Russia’s always had the challenge of not having great maritime access, just as a result of its geography, and so to the extent that it wants to be an active player in [the Mediterranean] … it has to have some ability to operate outside of its own coastal waters,” Mankoff explained. This ambition is enshrined in Russia’s new maritime strategy, detailed in the Maritime Doctrine of the Russian Federation 2020. The strategy places particular emphasis on the Atlantic Ocean due to “NATO expansion, the need to integrate Crimea and the Sevastopol naval base into the Russian economy, and to re-establish a permanent Russian Navy presence in the Mediterranean,” according to Russian Deputy Prime Minister Dmitry Rogozin. The doctrine also stresses the importance of the Arctic, given its mineral resources and the easy access it offers to both the Atlantic and Pacific.

All this isn’t to suggest that naval strategy is the primary motivation behind Russia’s interventions to support pro-Russian separatists in Ukraine and Assad in Syria. In the case of Syria, Putin has a track record of opposing Islamist movements like ISIS (in fact, that track record is one of the factors that brought him to power in the first place). Mankoff suggested that Russia’s Syria policy could be a mix of the personal and the political, saying, “If Putin believes that Assad is his guy and that he has a personal obligation to him, then that may play a role above and beyond what the professional diplomats and strategic thinkers believe is going on here.” Additionally, Mankoff argued that the Russian government might be deliberately trying to draw a comparison between its unflinching support of Assad and America’s brittle support of Hosni Mubarak in Egypt, which dissipated during the Arab Spring. [...]

Read the whole thing for embedded links and more. I've posted previously about Russian Geography and History, that broader context also explains a lot. You can read that post here:

Russia, geography and history