Sodium hydrogen sulfite is a salt, with the sodium atom having a positive charge and the hydrogen sulfite molecule having a negative charge. Like simpler salts such as sodium chloride, this compound is soluble in water. The polar charges on the water molecules are able to arrange themselves around the cation and anion, and collectively overwhelm and isolate each molecule, surrounding them in a "shell" of water molecules. This is the first step in the reaction.
NaHSO3 → Na+ + HSO3−
At this point you might presume that the bisulfite ion (HSO3−) would strip a hydrogen from a water molecule, making itself into the sulfurous acid molecule and producing an OH− ion. However, this produces a basic solution rather than an acidic one, and research suggests that the sulfurous acid molecule does not exist in solution anyway.
The only other option is to treat the bisulfite as an acid; its pKa is around 7, water is around 14. This means the water is more basic and therefore willing to accept a proton, which the bisulfite happens to have. This will produce a sulfur dioxide molecule with a -2 charge, and a hydronium ion, which has a very high acidity, thereby making the solution acidic.
HSO3− ⇌ SO32− + H+
H2O + H+ → H3O+