The space we live in has to be complete, with the "outside" not simply another spatial direction of the same geometry. You might postulate another orthogonal $w$ axis that works the same as $x, y, z$, and we're not able to move that way because everything is lined up at $ w=0$ now so nothing can get get beside you to push in that direction. There might be some field that attracts everything to that $ w $ position, and it might be very strong.
However, you don't have to stand beside something at $ w=$ 5inches for example to push in the $-w $ direction. The physics is not “closed” over the domain of the dimensions of the current arrangement of particles. Effects can operate at right angles to the participants. That is the general thing you see with cross products. Gyroscopes would produce torque in that direction, electromagnetic effects would have more right angles to reach out to.
We also have the problem of space and time getting mixed up. You can't move only within a line or plane since relativity will change the spacial axes. In fact, a combination of boosts (accelerations, thrusts) cannot be expressed as a single resulting boost vector, as we can with Euclidean geometry and Newton’s laws. You get a boost and a rotation when you try that with special relativity.
So being stuck to a 3D manifold (free in x,y,z but w constant) won’t work since the meaning of $ w=0$ will change depending on your motion, so (1) what am I really stuck to? Or (2) the universe will blink out of view if you move.
If forces and motion in the forbidden direction are corrected via an attractive force, that will cause the force that got turned around (via precession, magnets, whatever) to be apparently lost. Being stuck to a position doesn’t change the fact that the axis exists and that will change things from how we know them.
So, the spacetime we have is complete with the laws of physics only involving that 3+1 dimension spacetime. The “out there” must be something different.
The D-branes in M-theory (which were discovered by working through the math, not put into the theory because it was part of the idea) anchor the ends of a string so it vibrates like a guitar string. Imagine if the sound depended on whether the string was vibrating in the plane parallel to the face vs perpendicular to it, as well as different frequencies. Vibration modes of strings cause them to be whichever particle or force. If you took a guitar string playing its note with its pleasant sounding harmonics and timbre, and said that’s an electron, what happens if you unhook the ends from the instrument?
Well, you could restring it on a different instrument and if everything is the same, you have an electron in a different, disconnected universe.
You need to grab not just the objects from our universe but a piece of the substrate too, to keep them in their context. You can’t pick, up Mr. A. Square from flatland; you have to cut out the little patch of flatland that he's existing in.
So the higher space beings can do stuff to us. What happens analogously, if you stick your hand through Flatland’s plane? If the 4D hand is made up of string that had different D-brane endpoints and presented a 4D space to manifest the particles in, we’re heading to the idea that all metaspace is the same stuff and the universe and its dimensionality and laws are due to the specific strings present. How would electrons interact with the truly alien string it came into close proximity to? Let’s say that the beings engineered a string that manifests like an electron in profile and also interacted meaningfully with their native particles.
I see two approaches now. The classic idea of the higher-d object passing through and being perceived as a slice of the full object could be contrived by design. Instruments would pierce the lower-d universe and provide live interaction and a communication bridge.
The other approach would be to build a little patch of lower-d space filled with stuff, and graft it into the lower universe. That's the inverse of being able to pick up a patch with its contents.
But, how do they figure out what’s in the patch they sampled, if not with the probe method? Maybe the probe works with the sample in a large machine after modifying it: freezing and “staining” it. Now how would that affect contact? You would not meet A. Square in his home, but but destructively analyze him by blasting the sample apart in the LCLS !