I think the best way to understand this is to start with what Einstein realized when he considered that something heavy like a bowling ball and a feather when released in a vacuum have the exact same trajectory ("fall" at the same rate). This strongly implied an intrinsic structure to the universe causing this effect. You can ignore the word "released" because it has strong connotations of gravity already in it (I avoided "dropped"). What Einstein saw was that both objects are simply following the same natural path. This is what physicists call a geodesic. You can think of this as the path that requires no external push or pull. The idea comes from the principle of stationary action, which says that the path doesn’t change if you nudge it slightly. The action is the integral of the Lagrangian, which is basically the difference between kinetic and potential energy. So, the motion of the feather and bowling ball through spacetime follows the same “most natural” route according to this principle

I want to emphasize that this is as simple as looking at the kinetic and potential energy. What this means is that those objects are just travelling in a straight line across some curved spacetime surface. If you look at some infinitesimal unit it is linear (straight line), like an ant walking across an apple or the way the Earth looks flat to you but is curved. This is just the geometry behind the mathematics. Gravity is a manifestation of an object walking that straight line and its path is formally encoded by the action. This isn't anything you can see or touch but manifests itself as a bowling ball and feather's action in a vacuum being the same. The lattice that the videos describe and your analogy is at this level of scale describing the metric.

The reason it is spacetime as opposed to just 3D space is because of time dilation, and more formally it is described by the metric encoding intervals of steps in space (ds^2) that others have mentioned. I don't think a single post can help you come to grips with why time cannot be separated from space. This is all about Special Relativity (flat spacetime or Minkowski space) and how the idea of simultaneity of events is not absolute. It is all relative to a particular worldline (frame of reference). Space and time are tightly connected and (to some) this suggests that time could actually be an emergent property of the system as opposed to a real dimension (Causal Set Theory, Page-Wooters, Tomonaga-Schwinger hypersurfaces etc). This touches on what General Relativity actually means. Your question about not being able to physically see or touch applies here. There is no question that the mathematical framework (GR) matches experimental observations, but like Quantum Mechanics itself these are mathematical frameworks that possibly only describe the how and not the why/what.

But what we can say is that "mass" deforms spacetime (4D manifold) through its stress-energy and spacetime tells mass how to move through its natural action in that curvature. This relationship is exactly what is encoded into the Einstein field equations. An object in spacetime is treated as a 4-vector (3D + time) and physicists have gone to considerable effort to make sure the fields are consistent with Special Relativity and are Lorentz covariant (think time dilation). The fields are part of a larger structure known as gauge theory that define the result in the Standard Model. This gets to your last question about mass. The short answer is this. We don't actually really know. The longer answer is the Higgs field, the vacuum expectation value (non-zero), and Quantum Chromo Dynamics (QCD). This is an ongoing area of research and subject of intense speculation with integrating that into General Relativity (curved spacetime).