It's hard to know what to quote from this discussion, so I shall chicken out of doing so.
Historically, there was a profound change in understanding of the foundations of mathematics around 1900, especially with Hilbert and the rise of modern logic. We now set up theories which are purely sets of formulae and rules for manipulating them. Then we relate those to models, and consider various metalogical matters (consistency, completeness, etc.). But we can't understand how earlier mathematicians thought if we try to force their work into this modern framework.
The Greeks regarded geometrical objects as idealized physical objects. A line drawn with a pencil can be imagined getting thinner and thinner until it is an ideal line of zero thickness (Euclid I Def. 2). Geometrical objects can even be imagined as moved around and fitted on top of one another (Euclid I.4 proof), in which case they are equal (Euclid I Common Notion 4). It's easy to see why Plato liked geometry, because it gave such good examples of his theory of ideals.
But thinking that way carries the danger of relying too much on physical intuition when looking at actual diagrams. That seems to have been the trap which Euclid fell into in I.1, when he assumed that the two circles would intersect. A clear view of such things was achieved by Pasch in the 19th century. I think Euclid I.16 is an even more significant example. Near the end of the proof he claims that one angle is greater than another (Common Notion 5: "The whole is greater than the part"). This is just an intuition based on the diagram, and in elliptic geometry it is actually false.
However, remember that logically structured mathematics began in the 5th century B.C. (probably with Hippocrates), and was pursued by quite a small number of people in each generation. Just over a century later, those few dozen people had reached the level of Euclid's "Elements". Then in the 3rd century B.C. came the formidable achievements of Archimedes, not to mention Apollonius. Rather than emphasize the faults which we can see by modern hindsight, I prefer to admire the enormous achievement of the Greeks who reached such a high level in such a short time.