Problem: classical geometry is not happy with infinitesimals
Newton is systematically trying to avoid basing calculus on infinitesimal geometric quantities. We can see this from how he emphasizes that his method is consistent with the "ancient" standard of rigor:
To institute an Analysis after this manner in finite Quantities... is consonant to the Geometry of the Ancients... in the Method of Fluxions, there is no necessity of introducing Figures infinitely small into Geometry.
(Page 4 of the English translation cited in the question, page 6 of this scanned Latin edition - note that the page numbers refer to the numbering as printed on the actual scanned dead-tree pages, not the electronic numbering.)
Recall that before the modern era, the standard for rigor in mathematics was Euclid, and Newton followed the tradition by using geometric figures and arguments whenever possible. Now sure, classical geometry had what we now call the method of exhaustion. But it did not have the notion of limits or infinitesimals the way Newton did. Especially since the foundations of calculus were controversial from the start, Newton needed his ideas to be agreeable with the standards of classical geometry.
Solution: frame the foundation in terms of motion, not geometry
Calculus has both a geometric interpretation and a dynamic interpretation - think tangent lines vs. rates of change. You learn this early on. But arguments directly in terms of infinitesimal geometric figures don't fit into the classical geometry. If you appeal to motion and change, though, you can use perfectly reasonable and familiar concepts. For example, a secant line that moves and becomes a tangent line sounds a lot more believable than supposing you can just take two "infinitely close" points and take the secant line through them. There's some hand-waving going on either way, but the latter flies directly in the face of classical geometry, which, remember, was the standard of rigor. Framing calculus in terms of motion shifts the controversial foundation further from geometry, where it would face stricter scrutiny.
With this in mind, we can interpret the quote as establishing the emphasis on the dynamic viewpoint. Essentially, Newton is saying: Don't worry, we're not gonna do infinitesimal geometry. We're not gonna break a line or anything down into absurd infinitesimal parts which may not really be possible. Just view a line as the path traced by a moving point. With this viewpoint, we can appeal to concepts like velocity to justify our arguments. We can usefully apply calculus to geometry, without having overtly infinitesimal geometry.
And of course the perspective helps teach the important interplay between the geometric and dynamic interpretations of calculus.
P.S. Notes on the quote
For completeness, here is the quote in Latin:
Quantitates Mathematicas non ut ex partibus quam minimis constantes, sed ut motu continuo descriptas hic considero. Lineae describantur ac describendo generantur non per appositionem partium, sed per motum continuum punctorum; ...
You had some hypotheses about some particular words and phrases in the quote, so here are my comments on them specifically.
- You guessed that "mathematical quantities" ("Quantitates Mathematicas") means curves. It looks it just means quantities. For example, compare with page 5 of the Latin, page 4 of the English: "Fluat quantitas $x$ uniformiter & invenienda sit fluxio quantitatis $x^n$." ("Let the Quantity $x$ flow uniformly, and let it be proposed to find the fluxion of $x^n$.") Clearly $x$ and $x^n$ are numerical quantities.
- "Parts" and "points" are not the same word ("partium" vs. "punctorum"); in fact, Newton is contrasting the two. Parts are pieces you divide something into. Points are points, although since you're thinking of them as moving, you might say point particles. I'm sure there's more to say about what a point is according to Newton.