Intuitive interpretation:

A curve has an infinity of points. So many that the amount counteracts the abscence of dimensions, like an $0\times\infty$ undeterminacy.

More precisely, a curve still has no dimension transversally, but a finite (or infinite) dimension longitudinally.

You can think of it as the limit of a necklace of pearls (points of finite area) in contact, getting smaller and smaller but more and more numerous. In the end, an infinitely thin but continuous string remains.

You are confusing theory with parctise.
A point does not exist in the real universe.
Anything you draw will have some dimensions.

A curve is completely determined by two facts:

• Knowledge of all of the points lying on the curve
• Knowledge that the curve is drawn on the Euclidean plane

When it's said that a curve is made out of points, one really means to include in the latter fact too, or something similar (e.g. a topology or a metric on the collection of points).

There are more sophisticated geometric techniques (e.g. tangent spaces, halos, germs, stalks) that probe the "infinitesimal" shape of the curve at the point;

For example, studying the tangent space to the curve would indeed allow you to say that, at each point where the curve is smooth, it consists of an infinitesimal line.

But just to reinforce my initial point, the shape of that infinitesimal line can be ascertained simply by knowing the curve is being drawn on the plane along with which 'nearby' points lie on the curve.

You are confusing reality and the idealized world of maths. In an ideal world, a point has no dimensions. You could never actually draw a point. In reality, you can. You take a pencil, make a point on a piece of paper and say "that is the point". But the fact that you can see the point already means that is has some dimensions (the ink occupies a non-empty area in our real, three dimensional world). You have not actually drawn the point, but a representation of the (idealized) point.

Similar, if you draw a curve, you need to draw it with some thickness. This means you don't draw the actual idealized curve, but a real life representation of that curve.

Maths is founded in the real world, but goes one step further. Maths discovers the rules behind things, and abstracts them into an idealized world. This ideal world does not exist in reality, but it is nevertheless useful.

For example, infinity is a purely mathematical concept, yet it is tremendously useful for real-life applications, e.g. physics.

A point in an n-dimensional Euclidean space is that point which consists of n coordinates which give a distinct location of that point.

A function in Euclidean space essentially provides a rule for defining a set coordinates in space for which that function is true.

You are correct in that the physical realization of a collection of 0-dimensional points is nonsense to create a physical 1-D curve, however, it makes tons of sense when you learn a bit more math in topics like set and measure theory

As at least one other answer has mentioned, it takes an infinitude of points to make up a curve. (Or to make up any other interval, like the rather simple "curve" of the closed interval [0,1], for example.)

At least one other answer has also mentioned the notion of "measure," and that's probably the more appropriate way to think about the small-ness of a point: it's said to have "measure zero." If you've taken calculus, then you may recall that you can remove a finite number of points from the interval that you're integrating over, and it won't affect the value of the integral. Why? Because each point only has "measure zero," so its contribution to the total value of the integral is negligible, when compared to the infinitely-many contributions made by all the other points in the interval.

Don't forget, when you have a closed interval (or, once you get to topology-level stuff, a "compact" interval), those intervals will have infinitely-many points in them. Like take [0,1] again; there are infinitely many points in that interval. But cut it in half; say, to [0,1/2]. There's still infinitely many points in that interval. Cut it in half again, to [0,1/4]; still infinitely many points in that interval. You can keep cutting it in half as many times as you can imagine, and you'll still be looking at an interval with infinitely many points in there.

All this "infinitely many" stuff that you encounter in math; it may not seem to line up with the real world all that well, but it's necessary for doing calculus (because it's at the heart of the concept of continuity).

This article by Prof. Lawrence Spector helped me a lot and I think it addresses exactly what your confusion is. I'm not sure what the policy is on just posting links, but I don't think i would do a great job paraphrasing what he wrote, so here goes: http://www.themathpage.com/aCalc/apoint.htm

I would also highly recommend reading on the same site: http://www.themathpage.com/areal/real-numbers.htm, about the historical evolution of the real numbers. I think it'll clear up a lot of the questions you probably already have or will have.

Some other answers have already mentioned the distinction between the ideal geometric world (which we can describe by mathematical rules) and the real world (which our ideal world is intended to approximate).

However, an interesting point is that Euclid never talked about dimension but merely said:

[Euclid Book 1 Def 1] σημειον εστιν ου μερος ουθεν

[a] mark is that of which [there] [is] no part // A point is that which has no part.

It is amusing that we can force a set-theoretic interpretation onto this as:

A point is a non-empty set with no proper subset. // A point is a singleton.

This coincides with the notion that two different straight lines that intersect do so at a point.

If $A,B$ are distinct straight lines with an intersection, then $A \cap B$ is a point.

In more set-theoretic terms:

If $A,B$ are distinct straight lines such that $A \cap B \ne \varnothing$, then $A \cap B$ is a singleton.

Nice? Furthermore, a line can then be said to be a union of points!