Gravity as a force is determined by the mass of objects. Although it's no longer considered as precise as the theory of general relativity, Newton's law of universal gravitation still suffices in general for most purposes to help clarify how gravitational force between two objects occurs. (And we know there's merit to it, because Newton's law was used to determine the existence of a planet beyond Uranus in 1821, purely due to the fact that Uranus's observed orbit did not behave according to mathematical expectations. It was hypothesized that a sufficiently large planet existed beyond Uranus that was perturbing its orbit; lo and behold, Neptune was eventually discovered in 1846.)
In our particular case, 'up' is just a frame of reference we use relative to our position on Earth, because the mass of Earth overwhelms all other objects within our immediate vicinity. As such, the gravitational force exerted upon us by Earth 'pulls' us toward it. 'Anti-gravity', in this case, would be an inverted reaction between matter and antimatter, in the sense that gravitational interactions would not behave the same as they otherwise would. (In other words, it would not be a 'sideways force', but a lack of force altogether.)
If anti-gravity (as hypothesized for the purposes of the described experiment) existed, then antimatter would not be attracted by the mass of Earth (being made of matter), and would instead behave more or less randomly (since there aren't many objects made of pure anti-mass within the vicinity). However, based on the experiment, anti-matter behaved just like regular matter in terms of gravitational interaction, as it all was attracted in the direction of the greatest gravitational force within the proverbial neighborhood: namely, the Earth itself.