For most of those countries, it's a mixture of factors:
- Russian financial support, military support, or energy support (particularly with regard to Syria)
- Opposition to the West and intent to deliver diplomatic defeats or to not be seen by a domestic audience supporting the West (Bolivia, Venezuela, Cuba, Nicaragua voting together in Chavista style against the West)
- Allegiance to Russia in diplomatic matters (particularly Belarus, to a much lesser extent Armenia)
- Refusal to set a precedent regarding annexation (particularly Armenia)
The last bullet point begs extrapolation on Armenian policy toward the breakaway province of Nagorno-Karabakh, which is ethnic Armenians in an area that was considered Azerbaijan at the breakup of the Soviet Union. It's today governed as a non-recognized state from Stepanakert, but with lots of support from the Armenian government in Yerevan. The Yerevan government generally regards NK and the Stepanakert government as closely affiliated with Armenia. There are clear indications that Yerevan might like to annex Stepanakert despite an unresolved border dispute with the Azeris. In particular, NK would be a non-contiguous exclave of Armenia and would be an enclave wholly of Azerbaijan - absent some territorial conquest.
The Crimea resolution might be applicable to the Nagorno-Karabakh situation, as an example of annexing a group of co-ethnics or co-nationals in a neighbor's breakaway region, so Yerevan does not want to establish bad precedent. Also, it would be useful to have Russia on its diplomatic side in the dispute. Armenia has had to walk a diplomatic tightrope in the Caucasus, with Muslim opponent nations, the Russian giant, and the Georgian cousin all creating a multipolar situation with confused and crossways goals and allegiances.