Why are there Three Dimensions?

 Gimbal locking, plus relativity.  (Alternatively, gimbal locking plus holographic principle, if the idea that relativity implies 2+1 dimensions, rather than 3+1 dimensions, doesn't seem incredibly obvious to you.)

Locally, at least, all dimensions eventually collapse down to 2+1.

Assuming the mechanism I'm thinking of works - it's not quite gimbal locking, but it's kind of the same principle, in which two rotational dimensions become locally dependent - any finite number of spacial dimensions should reduce to two spacial dimensions in practice for the oriented matter, over enough time, at least locally.  I don't think the last two dimensions can lock, because you need at least one non-time orthogonal dimension (because some interactions create orthogonal forces), but my thoughts are very vague right now.

This explanation probably will not actually work in conventional physics, because the distribution of mass/energy doesn't work out nicely, as those extra dimensions never actually go away or become compact, they just become dependent; I do not believe you would get the inverse square law under these conditions.