Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?
As I said in my comment, you are in good company---in fact, the company of Dedekind himself! In a letter to Heinrich Weber, Dedekind says the following:
(...) I would advise that by [natural] number one understand not the class itself (...) but something new (corresponding to this class) which the mind creates. (...) This is precisely the same question that you raise at the end of your letter in connection with my theory of irrationals, where you say that the irrational number is nothing other than the cut itself, while I prefer to create something new (different from the cut) that corresponds to the cut and of which I prefer to say it brings forth, creates the cut. (Ewald, From Kant to Hilbert, vol. 2, p. 835)
So Dedekind himself preferred not to identify the real number with the cut, merely saying that the mind somehow creates the real number which then corresponds to the cut. This is, however, a little bit obscure, so it's not surprising that most mathematicians (such as Weber!) decided to ignore Dedekind's suggestion and simply identify the real number with the cut. The reasoning behind this identification is roughly the following.
We know that any Dedekind-complete ordered field is isomorphic to the field of the real numbers. In particular, this means that any construction or theorem carried out in the real numbers could be reproduced inside an arbitrary Dedekind-ordered field, and vice-versa, by simply using the isomorphism as a "translation" between the fields. Hence, it doesn't matter what the real numbers actually are; for mathematical purposes, even supposing that there is such a thing as the real numbers, anything that we wanted to do with them could also be accomplished in an arbitrary Dedekind-complete ordered field.
Thus, if we could show that the cuts themselves satisfy the axioms for being a Dedekind-complete ordered field, then we could dispense with the real numbers altogether and simply work with the cuts themselves. And, in fact, we can show that this is the case! One need only to show that, given two cuts, $X$ and $Y$, it's possible to define operations on them corresponding to the usual operations on the real numbers, such as addition and multiplication, and that after doing so these operations will satisfy the field axioms. It's not difficult to see that the obvious operations will yield the desired result (exercise!), though it is somewhat laborious. If you are interested in seeing a detailed verification, I recommend reading, say, Appendix A of Yiannis Moshovakis excellent book Notes on Set Theory, which contains a very thorough discussion of the matter.
The Dedekind cut splits $\mathbb Q$ in two subsets of rationals, all the ones smaller than the desired real, and all the ones larger.
These infinite subsets are used because a real might not be a rational, but can be approached arbitrarily closely by rationals. And by using infinitely many rationals, you can get closer and closer. (You need them all because there is no "closest" rational.)
For instance,
$$1<\frac{14}{10}<\frac{141}{100}<\frac{1414}{1000}<\frac{14142}{10000}\cdots<\sqrt2<\cdots<\frac{14143}{10000}<\frac{1415}{1000}<\frac{142}{100}<\frac{15}{10}<2$$
As the concept of reals can only be defined using already established concepts, the real is defined to be one of these sets of rationals.
If this approach seems contrived to you, remember that a rational is an infinite set of integer pairs $(kp,kq)$ where $p,q$ are relative primes.
From this definition, the basic operations (addition, multiplication, comparison...) on reals can be defined, by reasoning on the infinite subsets. But once the algebraic properties of these numbers are established, they can be manipulated as if they were "atomic" entities.
Have you seen the construction of the integers (from the natural numbers)? The integers are constructed as classes of equivalences of ordered pairs, which is also "weird". In order for you to begin to understand this process, start by thinking about those definitions as implementations, or models, of structures that we will (artificially, you can say, but that is irrelevant) show that behave like we expect them to behave in order for them to be called as such (integers, real numbers etc). Later you will see that this distinction is mostly psychological.
The bottom line is: They are clever ways to show the existence of objects which realize the structure we are idealizing. Dedekind cuts is a particularly clever example, as I'm sure you will eventually appreciate.
One way to start appreciating the cleverness behind this construction (and also dispel the negative feeling of artificiality and/or confusion) is by trying to define the real numbers by yourself. Be critical in such construction, and you will realize that a lot of your attempts will be (most likely) circular.