This was done as ELIPhD by Raychaudhuri.
Essentially for any given spacetime we can calculate out geodesics for any freely-falling object; it's just the path these objects follow through spacetime unless otherwise disturbed. Here we're interested in such objects that couple only to gravitation. These "test objects" do not radiate at all, not even when brought into contact with each other, and they don't absorb radiation. They don't attract electromagnetically, or feel electromagnetic attraction, and they don't feel such repulsion either. They also don't feel the weak or strong interactions. So they're always in free-fall -- always in geodesic motion -- because they can't "land" on anything.
We take one further step into fiction and prevent these test objects from generating curvature themselves. You can fill flat spacetime with them, and spacetime will stay flat. This is completely unphysical, but it's a handy property for exploring General Relativity.
If we put such an object into flat spacetime, we can use it to define a set of spacetime-filling extended Cartesian coordinates, where we add time to the Cartesian x, y, and z labels. We set things up so that the object is always at x=0, y=0, z=0, but can be found at t 0. The units are totally arbitrary. You can use SI units of seconds and metres, or seconds and light-seconds, or microseconds and furlongs: for our purposes it doesn't matter.
We can introduce another such object offset a bit, so that it is found initially at t=0,x=200,y=0,z=0. Again, the units are unimportant, it only matters that the second object is not at the same place as the first. This object is set up to always be at y=0,z=0.
In perfectly flat spacetime, these two objects, for t=anything, will be found at x=0 and x=200 respectively, and always at y=0, z=0.
They do not converge, ever, not in the past or in the future. They also do not diverge. The choice of coordinates doesn't matter any more than the choice of units; we could change the picture to keep the second object always at x=200, and the first will move from x=0. Or we can let them both wander back and forth along x, but with constant separation. But let's stick with our first choice of holding the first particle at the spatial origin at all times.
Now, what happens if we give the first object a little bit of stress-energy (you can think of that as mass in this setup)?
The geodesics generated now are not those of flat spacetime, but rather much closer to those of Schwarzschild. We have perturbed flat spacetime with the nonzero mass.
The first object, if we keep it always at x=0,y=0,z=0 now causes the second object to be on a new geodesic that is x != 200 at different times. Depending on the relationship between the "central mass" at the origin and distance x=200, the geodesic evolution of x for all t for the second object might look like an elliptical, circular, or hyperbolic trajectory [1].
If on the other hand we give both objects the same mass, we end up calculating out geodesics that focus. There will be at least one time t > 0 where the test objects will occupy the same point in spacetime, t=?,x=?,y=0,z=0. (This is called a "caustic").
Raychauduhri showed that caustics are highly generic[2]: you need electromagnetic repulsion (which means a global charge imbalance, which is not a feature of our universe); strong gravitational radiation (which is not a feature of our universe except perhaps in the extremely early universe); or a metric expansion of space (which is a feature of our universe, and leads to large volumes in which geodesics diverge, avoiding caustics, and small volumes in which geodesics converge such that caustics are only avoided by non-gravitational interactions).
This is the General Relativistic picture of masses attracting each other: objects follow geodesics unless shoved off them (by e.g. electromagnetic interaction), or until they "land" on something; in most physically plausible spacetimes there are generically intersecting geodesics and most things find themselves on one; and so close approaches, collisions, mergers, and so forth are practically inevitable.
Lastly, consider an https://en.wikipedia.org/wiki/Accelerometer . A calibrated one in free-fall anywhere should always report "0"; dropping the same out of an airplane should show a slight upwards acceleration imparted by collisions with the air, and then a big upwards one upon contact with the surface. These collisions with air molecules and water or ground molecules shove the falling accelerometer off its geodesic. An accelerometer resting on the ground or on the airplane will show an acceleration somewhere around 10 m/s^2 in SI units: it is being pushed off free-fall by interactions.
Two accelerometers freely-falling in flat spacetime will eventually collide with one another thanks to the focusing theorem. Only as they collide will the accelerometers show nonzero.
Finally, you can even experiment with this yourself: install https://phyphox.org/ on a modern smartphone and rest it on the floor, take it with you into an elevator, jump up and down, or throw it a long way (try not to break it, and try to avoid it rotating much while in the air) and you'll see that when in flight it registers a near-zero acceleration, but a substantial acceleration when in your hand as you wind up and throw, and a substantial acceleration when it lands. While in the air your phone is in practically-geodesic-motion.
It's this property of free-fall -- the absence of acceleration, even if one is orbiting or falling straight towards some massive object -- that is at the root of Einstein's gravitation, and which distinguishes it from Newton's gravity. It is formalized into the https://en.wikipedia.org/wiki/Equivalence_principle .
Although your thrown phone and the Earth are interacting gravitationally, neither the phone nor the planet feels a "pull" towards one another during the phone's flight, or during a parachutist's drop. The geodesics generated around the freely-falling Earth and (effectively) freely-falling phone just lead to greater radial motion by the phone.
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Definitely not ELI5:
[1] https://en.wikipedia.org/wiki/Hyperbolic_trajectory
[2] https://en.wikipedia.org/wiki/Raychaudhuri_equation#Focusing...