Take an antlike entity on a two dimensional surface, it is like an ant but it has only two dimensions moving within the (x,y) plane. One can pile up an infinity of (x,y) planes with ants in their surface when a z direction is hypothesized by an Einstein of an ant.
In this example the ant could disappear from its plane, with a perpendicular motion and appear on other planes.
If we hypothesize a fourth space dimension for our three dimensional universe , we could disappear into the perpendicular direction into another three dimensional volume.
They will be a certain amount to the "left," a certain amount "high," and a certain amount front or backward (etc). It's still length, width and depth.
The example of a two dimensional ant should make you understand that different surfaces should exist when adding the third dimension. For the three to four dimensions it is different volumes. And also funny shapes, part within our volume and part outside it would really be confusing us.
String theories need extra dimensions so as to be able to embed the standard model of particle physics into the vibrations of the strings. The standard model is an encapsulation of practically all the data we have for the quantum mechanical framework of particle physics.
As people do not in reality disappear into other extra dimensional volumes, theorists needed to make the extra dimensions very tiny, to agree with the experimental fact that only in fairy tales people disappear and reappear. The compactified
curled up extra dimensions still fulfill the need for embedding the standard model, and also the fact that no extra dimensions have been observed in all our particle experiments either.
In the example of the ant, if one compactified the third ,z, dimension, there would be no danger of it sliding around surfaces not existing in its original world, because it could not fit into the curled dimension.