Half reactions are usually composed of components of oxidation-reduction (redox) reaction where one element is losing electrons while another is gaining so they can come together into a compound. The half-reaction shows part of a reaction where the combining elements are charged and the product, from the redox, is neutral.
In this case, the H⁺ is carrying the positive charge because it is donating an electron to O²⁻, which is the reason the O²⁻ is carrying the negative charge. Note that we need 2 H⁺ for every one O²⁻. The O is being reduced while the H is being oxidized.
