Why does water split into ions?
The splitting described doesn't actually happen in bulk water. There is just the autoionization reaction:

$$\ce{H2O + H2O &lt;=&gt; H3O+ + OH-}$$

The net transfer of a proton from one water to another creates two ions. This is somewhat disfavored. Hence, the autoionization constant is quite small:

$$\mathrm{K}_{w} = \ce{[H3O+][OH-]} = 10^{-14}$$

Neutralization is a somewhat related process, whereby one species is able to create an excess of hydronium, and another is able to create an excess of hydroxide. This creates an imbalance in the autoionization equilibrium, and it rapidly shifts left to consume excesses of both ions.