In Euclid, real numbers are represented as lengths of segments, or at least one deals there with the notion of lengths of segments, and the ratios of such lengths. Basic geometric results deal with ratios of lengths of sides of triangles, and the condition of similarity of triangles is fundamental. In particular similarity can be defined as equality of angles, but the immediate corollary one wants is proportionality of side lengths. Since numbers are not available yet in Euclid he deals with pairs of segments and their ratios. However there is the problem of defining just what one means by the ratio of two segments. Hence arises the problem of defining carefully real numbers, since a real number is nothing but the ratio of two finite line segments. (If you have read Galileo, he actually deals with pairs of line segments, since he still has no algebraic notation for real numbers.)
Now Euclid has the notions of dividing a segment into a finite number of equal parts, hence he can make sense of the statement that the ratio of two segments is rational, i.e. some finite subdivision of one equals some finite subdivision of the other. But he knew that there exist pairs of incommensurable segments, i.e. such that no segments can be used to measure, i.e. subdivide, both of them. So in order to prove the theorem that two triangles with equal corresponding angles must have proportional sides, he had to devise a way to say that the ratios of all three pairs of sides are equal even when that ratio is not rational.
His solution was the method, attributed to Eudoxus, of infinite approximation by rational ratios, described in Definition 5, Book V, of the Elements. It essentially says that two ratios are equal if and only if every rational ratio which is less than one of them is also less than the other. For us it is only a small step to saying that this means a real number is determined by all rational numbers which are smaller than it is. Equivalently, choosing a point on the line, determines a separation of (the rational points of) that line into two parts, those points on one side and those on the other side of that separation.
The converse notion, that a real number is a separation of the real line, and that this separation is determined by the corresponding separation of the rational points, is exactly our modern notion of a real number as a Dedekind cut. So the Dedekind cut harks back all the way to essentially our first book of mathematics. Formulating it as a definition of real numbers, by Dedekind, may have required waiting for Georg Cantor to enunciate set theory, and the idea that mathematical objects should be defined as sets. At least Dedekind was said to have been an early appreciator of Cantor's ideas. For us it is hard to read Euclid and not think that Dedekind cuts are already there.
In my own view, this definition of real numbers is important more for this close link to Euclid, since cuts are very clumsy and hard to work with. (The details are worked out in Rudin and in Spivak's calculus book I believe.) A much more beautiful definition to me, is that of convergent sequences of rationals, modulo null sequences, i.e. the metric space completion of the rationals using the absolute value metric. But one way or the other, one must deal with real numbers as approximated by rationals. Infinite decimals is also a nice concrete approach that I had success teaching to a small bright special high school class long ago, some members of which became professional mathematicians.