Solvation is part of it (particularly for soaps and mineral spirits), but the main factors are oxidation and, as you stated, acid/pH. Oxidation removes organic stains (e.g. foods, mold-stains) and, depending on the strength, disinfects (e.g. Bleach). Mutually exclusive from oxidizing cleaners, acid-based cleaners dissolve insoluble inorganics (lime/rust build-up).