First, only a test particle "falls" along a geodesic. A test particle is an idealized object not only at rest, but which also does not itself contribute to the curvature of spacetime (no mass, no energy). An apple can be considered as a test particle in a system including earth, but it is a simplification, as the overall spacetime is determined by the dynamics of all objects that supposedly live "in" it; see this related answer of mine.
Now a massive object would also follow a geodesic, if we take this object into account in the spacetime itself by considering how it itself distorts spacetime. See this question - as John Rennie says there, it is a matter of terminology. The main point I want to make here is that spacetime is not a background. There is no "fabric" of spacetime.
Second, and as you correctly state in the question, a geodesic is not a purely spatial trajectory, it is a 4-dimensional curve, so it is actually misleading to think about "falling along" a geodesic, because the dynamical aspect of "falling" is already a part of the geodesic itself. A geodesic represents the world line of a test particle, and as such it is static: it tracks the object position in the most generalized sense, in all possible observer-relative decompositions in space and time of the unified spacetime it lives on. In other words it says where the particle is at any time in a way that is independent of any observer, an absolute way.
This means that "falling" along a geodesic is simply equivalent to "being somewhere" for a relativistic object. There is nothing forcing an object to follow a geodesic, a geodesic just tells when/where an object is, for its whole existence, an existence during which nothing ever messes with its position/momentum (nothing except gravity, but precisely because gravity is replaced by geometry in general reativity, it amounts to a non-interaction; see the answers to this question for a related discussion of the equivalence principle).
In short, a (timelike) geodesic is a geometric statement about inertia: where are you when you do nothing and nothing does anything to you? On a geodesic.