I'm not sure I fully understand.
To create a 5-cell, we fold 5 tetrahedral cells (similar to folding squares to make a cube).
Is the act of folding squares into a cube not a.. 3D act? For lack of a better phrase.
For instance, I'm not sure we can manipulate 1D the same as 2D? So to create a 4D 5-cell why can we apply a 3D technique?
Is there a more technical explanation hidden in the mapping? With 'folding' being an approximation? Could someone recommend a paper or literature to understand this?
Yes! Folding squares into a cube can't be done in $2$ dimensions, but requires a three-dimensional operation from your starting 2D shape.
Likewise, folding five tetrahedra together into a $5$-cell requires movement into the fourth dimension - the folding operation is not one you can perform in 3D.
You can picture this process as starting with a central tetrahedron $T$, with four others glued to it at its faces. As you fold it up, $T$ stays in the same place, while the outer vertices of the other four tetrahedra move into the fourth dimension. At the same time, the 3D projection of these vertices moves closer to the center of the original tetrahedron, although it is really "above" the center along this new axis.
As an analogy, you can consider the process of folding a tetrahedron from four triangles, as shown in this youtube video. If you were to look at the table from above, the points of the outer three triangles would converge on the center, but from 3D perspective you can tell that really this final vertex lies above the starting triangle.