How does a fabric containing 10% stretch material make it stretchy? Why should adding a small amount of a stretchy material make an otherwise non-stretchy fabric stretch?  Shouldn't the non-stretch fibres still constrain the maximum stretch of the fabric?
 A: It depends largely on the weave of the material and its components.  For example, you could add a small amount of stretchy material to these high performance sailing lines and it wouldn't change much because the non-stretchy components are still tightly bound together and dominate the structural characteristics.  Similarly, you can take a very stiff material like fiberglass cloth and stretch it in some directions -- you can stretch this stuff quite a bit if you grab it at the corners and pull since it just moves individual strands with respect to each other.  However if you grab at the sides and pull, parallel to the strands, it doesn't stretch at all.
You can make a fabric that stretches in any direction out of non-stretchy materials by weaving it in such a way that the non-stretchy components are somewhat "curly" and don't like in exactly a straight line.
The woven material is then stretchy because the stiff fibers can straighten out when the material is pulled on.  This can be done mostly in the design of the fabric -- a knit is much better than a weave for this.  For example this microscope image shows a knit fabric pattern that could probably stretch a lot, whereas this one probably can't stretch much at all. Cotton thread doesn't stretch much at all, but a cotton T-shirt (made from a knit fabric) can stretch a lot.
A small portion of a stretchy material to hold the non-stretchy fibers in place helps a lot too and opens up new possibilities for holding the non-stretchy fibers in place.   
A: Four years after the OP, this question still doesn't have an answer, yet it comes up in the first page of google searches on fabric stretch, so I will give it a shot.

Why should adding a small amount of a stretchy material make an
  otherwise non-stretchy fabric stretch? Shouldn't the non-stretch
  fibres still constrain the maximum stretch of the fabric?

You are completely correct in your speculation: adding stretchy material to an otherwise non-stretchy fabric does not increase the stretch of the fabric at all. As you suggested, the amount of stretch is completely determined by the length and alignment of the the least stretchy fibers in the weave or the knit. 
Instead, stretchiness is increased by making the least-stretchy fibers longer and distributing the extra length in some geometry that allows them to be pulled straighter when tension is applied in the direction of interest. The ability to distribute extra fiber using many different geometries is one of the primary contributions of knitted structures compared to woven ones.   
There are two reasons why high-stretch fabrics contain added elastic material.  The most important purpose is to improve the recovery of the fabric when the tension is released. The same knitted or woven fabric without the elastomer would stretch just as much, but would be baggy and would recover very slowly, or may not recover until processed in some way (washing, steaming, ironing).  Adding elastomer in the right amounts and the right places allows that same fabric to spring back immediately.  
The second way in which elastomers improve stretchable fabrics is by adding resistance to stretch.  Where a stretchable fabric without elastomers may  be stretched with very little force, adding elastomers increases the amount of tension required to stretch the fabric. This allows the fabric to retain its shape and structure until a designed-in amount of tension is applied. 
A: Google brought me here.  It failed to answer my question.  I might be able to add a good two cents here.
Not for nothing but all jeans are stretch jeans because they are cut on the bias/diagonal.  They would be stiff if cut like the average dress shirt or slacks. Denim leggings with 3% spandex are much stretchier than heavy denim jeans with 5% spandex.
Thickness of fabric and pattern layout are pretty big factors when it comes to stretch. 
If the cotton threads in stretch denim are anything like cotton yarn with elastic, then the cotton is spun and bunched up around the spandex in order to allow for stretch.  The cotton is pretty rigid.  That limits the stretch to add stability and durability.
So jeans or leggings would have stretch in one direction courtesy of cutting on the bias, and stretch in the other courtesy of spandex, with stability and durability added from the cotton.  
BTW cutting woven fabric on the bias adds stretch and drape regardless of material.  
edit - I wanted to add that knitting does not automatically create stretch.  Cotton is not automatically stretchy when knit!  Heavier cotton threads and yarns create garments that are super drapey when knit.  The reason a 100% cotton t-shirt is stretchy is because it made of incredibly fine thread that is knit tightly on a machine.  The same result would not come from crochet thread hand knit with size 2 needles.  So size plays a role too.
