What does a higher half cell potential mean?
That's not quite correct.

Half-reaction potentials are only really meaningful when you complete the redox pair, that is, you must have an oxidation and reduction reaction. The sum of the potentials is the potential for the full redox reaction at standard concentration and state. In that sense, the more positive either of the potentials is, the more favorable the overall reaction will be.

There is no preference for a neutral species in general. If you put sodium in water, the electron is quickly liberated and the end products of hydrogen and sodium ion are much more favored than the reactants of water and sodium.