How do curves consist of points?

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.

Euclid did say that that "A point is that which has no part". It is nothing more that a whisper of an indication that "You are here." You can't really see a point since there is nothing there.

"...if point has no dimensions, i.e. in other words there is nothing,
 then how is it possible to draw any curve?"

If you're thinking in terms of "connecting the dots", it isn't physically possible. Pick a ridiculously small positive number, $\delta$. It doesn't matter how small. It is a fact that there are as many points in the interval $(0, \delta)$ as there are points in the universe. There is no way you can physically enumerate all of the points in the smallest of curves. I don't believe that Euclid had our understanding of infinity, but I think he was aware of its paradoxical abundance.

But you don't have to draw curves. They are just a set of points. A graph is just a representation of that set and only serves to fuel our intuition.

Euclid's definitions of point, line, and segments, have zero functionality. They may sound nice but they don't really say anything useful.

If you want to know what a point really is, propositions like the following are much more useful.

"Two distinct points determine a unique line."

"If two distinct lines intersect, then they intersect at a single point."

addendum

Having reread your question for the upteenth time, it occurs to me that you may be thinking of infinitesimals. Infinitesimals are basically numbers that are smaller in magnitude than any real number but are not equal to zero. You may want to check out THIS and THIS.