How does diffraction occur in a reflecting telescope? It's understandable how diffraction occurs in refracting telescopes, because the lens itself is a small aperture for light to pass through.

But what about reflecting telescopes? Since light is reflected, and a mirror is not exactly an 'aperture'? How can light diffract in a reflecting telescope?
Diffraction has nothing to do with whether you are using a lens or a mirror in your telescope. Diffraction occurs for the same reason in reflecting telescopes as it does in refracting telescopes: it is passing through an aperture.  

There is no threshold aperture diameter for which you have diffraction. No matter how big the aperture, you will have diffraction. No matter the shape of the aperture, you will also have diffraction. This happens on any aperture. 

However, the bigger the aperture, the smaller the diffraction pattern, or Airy disc. In 
page 19 of the document you mention in the comments, there is a relation between the size of the aperture and the size of the Airy disc. 

To clarify what is said in the first paragraph of page 18


  If the telescope is replaced by one with a narrower objective, the images of the two stars would overlap too much and the observer would not be able to see them as separate stars.


If the two stars are replaced with two stars that are close enough together, the images of the two stars would overlap too much and in this case the observer would not be able to see them as separate stars either.



As a side note, refracting telescopes aren't necessarily smaller than reflecting telescopes. There are refracting telescopes with a lens more than a meter in diameter.  A lot bigger than most amateur reflecting telescopes !