What forces keep a cotton fiber together?
A glucose molecule is roughly 1 nanometer long (ref).  There are roughly 9,000-15,000 glucose molecules in a cellulose fiber (ref, see Table 5).  If we use 10,000 as the number of glucose units, then the length of a cellulose fiber will be 
$$10^{-9}~ \mathrm{m/molecule} \cdot 10^4 ~\mathrm{molecules/fiber}=10^{-5}~\mathrm{m/fiber} $$
or ~10 microns/fiber.

The mechanical strength of the fiber comes from two sources.


Structural: As your diagram illustrates there are many walls in the cellulose fiber.  All of these walls reinforce one another and build up the mechanical stability of the fiber. Each wall is composed of highly-organized cellulose chains.  "Highly-organized" is another way of saying "crystalline".  Cotton has an average crystallinity around 75%.  This allows the cellulose fibers to pack closely together further increasing mechanical stability.  The cellulose fibers in the different walls are orientated in different directions, again serving to increase the fiber strength.
Chemical: Here is a drawing of 2 glucose molecules from a cellulose chain.




Note that each glucose molecule has 3 equatorial hydroxyl groups oriented out and away from the polymer chain.  Because of the high crystallinity mentioned above, the cellulose chains are positioned very close to one another.  This means that thousands of hydrogen bonds can form between hydroxyl groups on different chains.  As a rule of thumb, a hydrogen bond might have a strength around 5 kcal/m; if we have thousands of these bonds holding the cellulose chains together, then we have the equivalent of hundreds of normal chemical bonds holding the chains together!  These hydrogen bonds account for much of the stability of the cotton fiber.