Here's a bit about Kaluza-Klein extra dimensions, as one finds in string theory. K-K theory started with just one extra dimension in an effort to unify gravitation and electromagnetism.

https://plus.maths.org/content/10-dimensions-and-more-string...

I'll try to generalize and simplify a bit:

There are other types of theory with extra spacetime dimensions, too. A physical interpretation (for many of them, anyway) about why we don't see the extra dimensions is that we can't move in them at all, or only very very little.

Consider what you're doing right now, sitting in place at your computer. In a standard 3-coordinate system of your room, with the origin of Cartesian coordinates (x,y,z) = (0,0,0) the floor beneath the centre of your chair, and measuring distances in light-nanoseconds, your centre of mass is a couple light nanoseconds in the z direction, so let's label that (0,0,3). We can extend from Cartesian (spatial) 3-coordinates to Minkowskian (spacetime) 4-coordinates by extending (x,y,z) to (x,y,z,t). In that case, we might decide your centre of mass right now compared to an extension of our previous origin is (0,0,3,3 * 10^10), where we measure t in nanoseconds rather than light-nanoseconds[1]. If there is a fifth coordinate, and it's spacelike, we'd measure in light-nanoseconds and then you're now at (0,0,3,4 * 10^10,0), because you're still not moving in space, only in time.

If you were to raise or lower your chair, z would increase or decrease respectively, t would continue to tick along in nanoseconds, and all the other dimensions would remain 0. Likewise, if you slide your chair to the right, you'd increase the x dimension so now you could be (3,0,3,6 * 10^10,0).

You can keep moving arbitrarily within this local system of coordinates, and try as hard as you like, you would not be able to go backwards in the time coordinate, and you would not be able to move in either direction in the extra spacelike dimension.

In theories that import extra spacelike dimension theories typically allow for extremely small movements through them, usually as a small oscillation, say (using nano light seconds still) 0->10^-35->0. That's so small you won't notice, and it usually happens across a tiny interval along the timelike dimension (e.g., you're only nonzero for a tiny fraction of a nanosecond). Moreover, even such a small oscillation is only allowed under extreme (usually high) energy conditions.

Kaluza-Klein's original theory represented this sort of minor deviation from zero as a loop with a circumference on the order of 10^-33 cm, compared to the line-like (uncurled/unlooped) other axes which extend in principle to infinity. [2]

There's the opposite form of inaccessibility, where everything in the universe that we can observe propagates identically along the extra dimension as everything else, so for example you can't see the difference in the "w" dimension between you and Voyager 2, even though you can certainly see differences in the other four dimensions. [3] For an example of how this could be useful, one can write down a theory that in such a background a very weakly-interacting particle (like a graviton or a neutrino-like particle) might propagate differently from everything else.

Finally, from various precision tests of inverse square laws, and careful observation of the orbits of various massive bodies at many different mass scales, we can make some pretty strong bounds on how accessible small and large dimensions are. They're basically inaccessible to humans at present level of technology. Consequently, it is still entirely sufficient to use 3 spatial and 1 timelike dimensions, and safest to assume (thanks to the weight of a lot of evidence) that that's all there is. If high energy particle physics or the study of extreme gravitational physics provides evidence that 3+1 is insufficient, that'll be really cool for theoreticians. But that's a huuuuuge if.

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[1] ignoring issues of sign

[2] this assumes a flat spacetime highly similar to Special Relativity; importantly KK theory deliberately breaks that similarity, but this conceptual comparison of the shape of the five axes will have to do unless I can think of some better way of putting it non-mathematically

[3] ignoring general relativistic effects