To really understand this you should study the differential geometry of geodesics in curved spacetimes. I'll try to provide a simplified explanation.
Even objects "at rest" (in a given reference frame) are actually moving through spacetime, because spacetime is not just space, but also time: apple is "getting older" - moving through time. The "velocity" through spacetime is called a four-velocity and it is always equal to the speed of light. Spacetime in gravitation field is curved, so the time axis (in simple terms) is no longer orthogonal to the space axes. The apple moving first only in the time direction (i.e. at rest in space) starts accelerating in space thanks to the curvature (the "mixing" of the space and time axes) - the velocity in time becomes velocity in space. The acceleration happens because the time flows slower when the gravitational potential is decreasing. Apple is moving deeper into the graviational field, thus its velocity in the "time direction" is changing (as time gets slower and slower). The four-velocity is conserved (always equal to the speed of light), so the object must accelerate in space. This acceleration has the direction of decreasing gravitational gradient.
Edit - based on the comments I decided to clarify what the four-velocity is:
4-velocity is a four-vector, i.e. a vector with 4 components. The first component is the "speed through time" (how much of the coordinate time elapses per 1 unit of proper time). The remaining 3 components are the classical velocity vector (speed in the 3 spatial directions).
$$ U=\left(c\frac{dt}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau}\right) $$
When you observe the apple in its rest frame (the apple is at rest - zero spatial velocity), the whole 4-velocity is in the "speed through time". It is because in the rest frame the coordinate time equals the proper time, so $\frac{dt}{d\tau} = 1$.
When you observe the apple from some other reference frame, where the apple is moving at some speed, the coordinate time is no longer equal to the proper time. The time dilation causes that there is less proper time measured by the apple than the elapsed coordinate time (the time of the apple is slower than the time in the reference frame from which we are observing the apple). So in this frame, the "speed through time" of the apple is more than the speed of light ($\frac{dt}{d\tau} > 1$), but the speed through space is also increasing.
The magnitude of the 4-velocity always equals c, because it is an invariant (it does not depend on the choice of the reference frame). It is defined as:
$$ \left\|U\right\| =\sqrt[2]{c^2\left(\frac{dt}{d\tau}\right)^2-\left(\frac{dx}{d\tau}\right)^2-\left(\frac{dy}{d\tau}\right)^2-\left(\frac{dz}{d\tau}\right)^2} $$
Notice the minus signs in the expression - these come from the Minkowski metric. The components of the 4-velocity can change when you switch from one reference frame to another, but the magnitude stays unchanged (all the changes in components "cancel out" in the magnitude).
I am afraid your allegoric figuring out why gravity causes time dilation is further confusing. The physical reality is measured by the metric of a reference frame which composes the coordinates, but the single coordinates do not have necessarily a specific meaning in GR (general relativity).
The Einstein equivalence principle allows to describe gravity in terms of geometry (metric) of a curved spacetime.
If we consider a static (Schwarzschild) spherical mass, radially the curvature is different and the proper time of stationary observers, as given by the metric, is progressively slowing as you approach the mass, if compared to the time measured by an observer far away from the mass.
The time dilation of SR (special relativity) is still a different concept as it is symmetrical between two observers in uniform relative motion. Instead the gravitational time dilation is not symmetrical; in fact the stationary observer measures a time contraction if compared to the far away observer.
Best Answer
To understand this consider the analogy of two cars driving north from the equator as discussed in the question: Why does the speed of an object affect its path if gravity is warped spacetime? The relevant diagram from that question is:
Due to the curvature of the Earth's surface the two cars converge even though they started out parallel, and this is what happens in general relativity, except that there we need to treat time as a dimension as well.
If you look at the diagram it should be obvious that the faster the cars drive north the faster they will converge i.e. the large the apparent force between them. In fact as discussed in Geodesic devation on a two sphere if the cars start a distance $d$ apart their separation at time $t$ is given by:
$$ s(t) = d \cos\left(\frac{vt}{r} \right) $$
and a quick differentiation later we find the acceleration $d^2s/dt^2$ is proportional to $v^2$. So the apparent force between the two cars is proportional to the square of their speed northwards.
In general relativity we find for a stationary object the four acceleration is given by:
$$ {d^2 x^\mu \over d\tau^2} = - \Gamma^\mu_{\alpha\beta} u^\alpha u^\beta $$
See How does "curved space" explain gravitational attraction? for where this equation comes from. The quantity $u$ is the four velocity, so once again we have a quadratic dependence of the acceleration on velocity just as in the case of the 2-sphere above.
And this neatly explains why even a small curvature causes large accelerations. It's because the magnitude of the four velocity is always equal to $c$, so in the equation for the acceleration we are multiplying the (small) curvature term $\Gamma^\mu_{\alpha\beta}$ by the (very large) term $c^2$. So even at the Earth's surface, where the spacetime curvature is barely measurable, we still get an acceleration of $9.81~\textrm{m}^2$ because we are multiplying that small curvature by $9 \times 10^{18}$.
As for time passing, no it does not imply time flows. In GR the trajectory of an object is a line in a 4D manifold i.e. the set of all possible positions in spacetime of the object. While humans perceive the position on that line to be changing with time there is no equivalent concept of flow in GR. For more on this see What is time, does it flow, and if so what defines its direction?