I don't know why almost no one provided actual viable reasons and only shared some of their personal opinion (e.g., "only the old guys like Russia") without trying to answer the actual question from OP. To understand this connection between Serbia and Russia you need to look at the broader historical and cultural context. Keep in my mind that I'm is Russian myself, so I may be a little bit biased:
-the Balkans are a mix of different religions, including Islam and different branches of Christianity. Some countries are Muslim, most are Catholic and the minority are Orthodox Christian. This by nature makes some Catholic countries closer to their Catholic neighbors, Orthodox to Orthodox neighbors, etc.
-add to that several hundreds of years being under Ottoman Empire rule with religious persecution and several attempts to revolt from various Balkan nations;
-Serbs being Orthodox Christians, naturally grew closer to Russia being one of the main Orthodox superpowers at the time. Especially considering - Russia had a lot of geopolitical clashes with Ottomans and used this Orthodox connection to promote internal conflicts in this region in order "to protect Orthodox population";
-for example, Russia even lobbied the creation and sovereignty of modern-day Bulgaria (another Orthodox state in the Balkans) as a must-have condition after one of those Russian-Turko wars in the San Stefano treaty of 1878;
-all of that sort of historically created "little bro-elder bro" relations between not only Serbia, but other Orthodox countries in the region. If some of those relationships to some reasons worsened throughout time (e.g., Bulgaria quite famously opposing Russia in both World Wars lmao), relationships between Serbia and Russia kept strong throughout those years;
-even the sole reason (at least formally) of Russian Empire entering the WW1 was "to protect our orthodox brothers in Serbia". In reality ofc that was a major conflict between superpowers for changing power balance in the world. But in terms of formal reasons - that was the main one that was used for the population and the army. So Serbs also remember that;
-as for the various comments here telling that "Russia didn't help Serbia during the Kosovo conflict" - that's also maybe not a blatant lie, but certainly doesn't provide most of the truth. Russia consistently backed up (and continues even now) Serbia in this conflict diplomatically, being one of the most active countries in the UN in stating that that was the "violation of international law", opposing all the military actions against Serbs, etc.
-Russia still doesn't recognize Kosovo as an independent state and doesn't have any diplomatic relationships with them. As for the military support - you need to understand that 1999 Russia wasn't the USSR or even modern-day Russia, it was pretty much a weak nation with lots of internal conflicts, crumbled economy after the 1998 financial crisis, banditism, etc. - so obv it didn't have the capacity to oppose US and European superpowers. Despite that, there were weak attempts to be involved (Pristina airport), but that wasn't effective

All of that created strong cultural and historical ties. Most of them are based on Orthodox Christianity. There are historically strong connections between Russian, Serbian and Greek football fan groups (e.g., Spartak Moscow-Red Star-Olympiacos and CSKA Moscow-Partizan-PAOK) that are based on the "same colours-same faith-same team" principle. To say that only elder people and delulus support Russia in Serbia maybe is a Reddit thing, but at least in my personal experience I've met lots of younger Serbs in real life, who would instantly get delighted once they would hear that I'm Russian, invite to get a drink and would shout some "Срби и Руси браћа заувек!".

I understand that maybe that isn't as popular between youngsters both in Russia and Serbia, especially considering the protests against Vucic and all that stuff, but it still exists. But to say that it completely doesn't exist - is not true at all. Perhaps to get the answer to that question OP need to talk to some people in real life, since Reddit is a quite niche audience with all due respect.