Why will decreasing my sodium intake help with my meniere's-related vertigo?

Absolutely. Salt, fat, caffeine, alcohol and tobacco are the enemy when you have menieres disease. The better you watch your diet the better your symptoms will be.
Fluid overload. Water follows salt. The more salt, or sodium, there is in your tissues, the more fluid you retain. Meniere's disease is a condition of fluid overload in the inner ear to an extreme which we call "hydrops" and it injures the nerves of balance and hearing.
It may. Meniere's disease is a chronic, progressive form of dizziness. It usually has 4 symptoms associated with it: 1) hearing loss ( begins as low frequency) 2) dizziness (many times incapacitating) 3) tinnitus 4) pressure in the ear or head. First line treatment is sodium restriction/ diet modification followed by a diuretic. In extreme cases surgery may be indicated.