Why does convex lens diameter affect its magnification? Usually , reflector telescopes will have a greater angular magnification because the focal length can be increased in the same physical distance, so to achieve the same angular magnification, refractor telescopes (with lenses, no mirrors) would require a 'large diameter and long focal length' for the same angular magnification.
This comes from page 15 https://www.studocu.com/en-gb/document/cardiff-university/foundations-of-physical-chemistry/other/oxo-aphy-q-01uu-op01-xxaann/4280291/view 
How and why does the lens diameter affect the magnification produced by a convex lens? I'm looking more for an intuitive explanation.
 A: The lens diameter improves the resolving power of a telescope. A telescope does magnify, but more importantly, a telescope resolves- It makes it easier to view two distinct points as separate points. The objective does the resolving job, and the eyepiece does the magnifying job.
Resolving power of a telescope (or lens, for that matter) is defined as the reciprocal of the smallest angle subtended at the objective lens of the telescope by two different points which can be just distinguished as separate.
Mathematically, it is given by the expression 
$$Z=\frac{a}{1.22\lambda}$$
You can obviously see why the aperture size ($a$) should be quite large. As I had stated, a telescope is not designed to magnify, but to resolve. And the eyepiece takes the tiny resolved image (very tiny), and makes it (magnifies) large.
Since you ask for a more intuitive explanation, I'll try to answer it here.
Suppose you are in a helicopter flying above a fair. You will see a huge mass of people, and if the helicopter is flying very high, then you will not be able to distinguish two individuals amidst the crowd. 
As you move nearer, it becomes easier to do so - assume that no one is touching each other and everyone is 1m away from one another. Technically, since everyone is separate, whatever may be your height, you should be able to see them distinctly. But you don't because of a phenomenon called diffraction.
Because of diffraction, what should appear as a point appears rather like an archery target. Two such sources would have their diffraction Maximas almost coinciding. For a larger aperture, these coinciding region can be reduced, and we can get a better-resolved picture. 
It might not be correct to say that a telescope magnifies-the image is actually smaller than the object. The telescope brings an object that is far away, to a closer point.  
