Why are there H+ ions in water?

You ask an interesting question that deserves a complicated answer :)
The traditional and simple answer to this question would be that molecules such as water can spontaneously dissociate in solution due to thermal energy and that the molecule of water is therefore in equilibrium with its ions
Of course, no reasonable chemist would ever tell you that H+ ions exist in water, because well … they don’t. Even without going into the depths of quantum mechanics. It is fairly obvious that the H+ ion is a mere proton. This kind of charge density makes it an extremely hot target for the solvent water molecules (which are around in plenty). Therefore, upon creation, the proton forms the solvated species H30+ which is the species that can be said to exist in water.
Saying that the H+ ion exists in water is just a way to explain the pH and pOH of water to beginner chemists.
However, I should tell you even the above description is a fairly crude one. The pH of water is also dependent on quantum mechanical effects such as shielding. Which is the fact that the electron (which is quantum mechanically, a cloud) is more around the oxygen nucleus (due to a lot of factors like electro-negativity) and due to the fact that the only electron which hydrogen has, has spread towards oxygen. The other side of the hydrogen is now vulnerable to attack. This makes Hydrogen tempted to leave behind its electron and grab hold of electrons from other molecules giving it typical Lewis-acid type behavior.

You might want to look up bonding in chemicals (the LCAO approach) for further detail:
http://en.wikipedia.org/wiki/Chemical_bond#Covalent_bond
http://en.wikipedia.org/wiki/Molecular_orbital_theory
http://en.wikipedia.org/wiki/Valence_bond_theory