On the model of Halliday's Systemic Functional Linguistic Theory, the three dimensions of space can be understood as circumstances of the unfolding of processes: location and extent (the distance or interval between locations).
On the model of Einstein's General Theory of Relativity, intervals of space, the distance between two locations, are not uniform across the Universe, but are either relatively expanded or contracted. On the one hand, the cosmological expansion since the Big Bang is the ongoing relative expanding of spatial intervals, the distance between locations. On the other hand, the presence of matter has the effect of relatively contracting spatial intervals, with the distance between locations decreasing with relative proximity to the centre of mass. (The limiting case is the singularity of a black hole, which is the contraction of spatial intervals to 0.)
The constant acceleration of a falling body due to gravity is explained by the constant contraction of spatial intervals with proximity to the centre of mass of the two bodies. As spatial intervals contract, more and more of them are traversed by the falling body, such that the distance travelled for a given unit of time — the velocity of the object —increases at a constant rate.
The direction of gravitational attraction to the centre of mass of the two bodies is explained by the fact that, unless acted upon by an external force, a moving body always takes the shortest trajectory between two locations: the geodesic. Because spatial intervals are contracted in the direction of the centre of mass of the two bodies, the shortest distance, for each spatial interval, is in the direction of that centre of mass.
The trajectory of an orbiting satellite is perpendicular to the direction of gravitational attraction, but the shortest spatial intervals are always in the direction of the centre of mass of the two bodies, and it is this that accounts for its curved trajectory. That is, gravity is not the curvature of space, but the contraction of space that results in curved trajectories of bodies through space.
(In the 'rubber sheet' model that is used to explain gravity, the relative contraction of spatial intervals is misleadingly represented as a deepening deformation of the sheet — that is, as an additional dimension instead of variation in the dimensions of space.)