Obviously some, like the conservative military establishment and the cossacks, had direct personal stakes in the outcome of the war, and many were pressed into service against their will, but from where did the White movement derive its popular support and what were the hot-button issues and rallying cries that led ordinary Russians to fight for the many myriad warlords and armies of the White Army?
The drum is thundering and thundering...For what?...
When you rise to your full height, you,
giving to them his life,
when you throw a question to their face:
what are we fighting for?"
Mayakovsky, "Call to Account" (K otvetu!)
There are many reasons why an average Russian might support the White Army, which was far more ideologically fractured than the Bolsheviks. As you not there are certainly classes that are deeply bound to the former Empire, while in 1917 - or 1919 - it is far from clear just what will happen in the Civil War. It's easy to imagine, I think, why the comparative stability of the late Empire - with all its faults - might be more alluring than the chaos following the Revolution. There is no guarantee that the Bolshevik regime will be any better than the Tsarist, though here the comparison is actually with Deniken and Yudenich and Kolchak (this, of course, is one of the reasons the Whites struggled - there isn't actually a singular army, and certainly nothing close to a singular vision, for Russia). We might also consider the Bolshevik view of religion, the very real crimes of the Bolshevik revolutionaries, the Bolshevik vision for industrialization. Small landholders far outweigh the urban poor that serve as the initial core of the Red Army.
Beyond the broad, general reasons - religion, culture, economy - personal literary pieces come to mind: Bulgakov's Days of the Turbins, earlier serialized as The White Guard. Bulgakov presents the personal reasons why one might support the Whites, a tragic remembrance (or idealization) of the past, which Stalin was reportedly engrossed by while Lunacharsky criticized the play as apologetic.
Alexander Blok's "The Twelve" is an early, complicated poem, where a prostitute is executed by the Bolsheviks on the snowy streets of Petersburg. The Revolution is certainly the death of Imperial Russia - in fact, Bely and Voloshin both compare Petersburg to Golgotha.