Imagine for a minute that the Earth is perfectly spherical, that sunlight arrives "from infinity", and that any given spot is either lit or unlit. Then, at any given time, exactly half the surface is in "daytime" and the other half in "night-time". The line separating the two is a Great Circle - a circle with the same centre and radius as the solid sphere.
It should be clear that any other Great Circle on this ideal Earth must intersect this penumbral line and be divided exactly into two equal parts of light and dark.
Now consider the path that any given place on the sphere travels as it rotates. It will move around a line of latitude (i.e. always eastwards). This path forms a circle parallel to the Equator. There is a line of latitude that is a Great Circle, and that is the Equator itself.
So we can deduce that a point on the Equator, in ideal conditions, will spend exactly half the day on the light side and half the day on the dark side of the Earth (except for the case we ruled out where the Sun is exactly over one of the poles).