Since breeden has already given you fine hints for proving the result is a Dedekind cut (once the statement has been fixed), I'll address the issue of the intuition behind it.
A Dedekind cut is just a way of breaking up the rationals into two complementary sets. Imagine the rationals arrayed in the real line: a Dedekind cut is like picking a point on the line, and looking at all the rationals that are smaller than the point, excluding the point if the point happens to be a rational.
The intuition is that we want to use Dedekind cuts to represent the real numbers. If all we know are the rational numbers, how do we describe a real number? A real number $r$ is uniquely determined by the collection of rationals $q$ with $q\lt r$, in the sense that
$$\{q\in\mathbb{Q}\mid q\lt r\} = \{q\in\mathbb{Q} \mid q\lt s\}$$
if and only if $r=s$. Moreover,
$$\{q\in\mathbb{Q}\mid q\lt r\} \subseteq \{q\in\mathbb{Q}\mid q\lt s\}$$
if and only if $r\lt s$.
But that's cheating: we are actually using the reals, and we are supposed to be trying to construct them with only rationals as reference. So we try to identify which intrinsic properties these sets of "all rationals less than or equal to a particular real number" have, and use them to define these sets. Then we can use the sets as proxies for the real numbers.
The three properties you give are an intrinsic way of describing what "the set of all rationals less than or equal to a particular real" would be: if they contains a particular rational $r$, then they contain all rationals smaller than $r$ (property 1); they are bounded above in the rationals (property 2); and they have no largest element (property 3).
This allows us to use these sets as stand-ins for the reals, without having to actually define the reals: we can just work with sets of rationals. This makes the order properties of the reals very easy (as I noted above): it is easy to compare these sets of rationals in a way that maintains our intuitive notion of "order" for the reals they are proxies for. But it makes the additive and multiplicative structures a little more difficult: we need to find a way of "adding" and "multiplying" these sets in a way that will correspond to what we want "addition" and "multiplication" of reals to be.
Addition is pretty easy: to get all rationals less than $r+s$ from all rationals less than $r$ and all rationals less than $s$ you just add them pairwise. So we can define the sum of two Dedekind cuts $D$ and $E$ as
$$D+E = \{ q_1+q_2 \mid q_1\in D\text{ and }q_2\in E\}.$$
And this works.
What you have here is part of the method to define "multiplication." If we just try to do the same thing we are going to fail, because $D$ and $E$ will have arbitrarily large negative elements, and when we take their product we will get arbitrarily large positive rationals, so we will not get a Dedekind cut (it will not be bounded above). So we need to do something more clever.
Intuitively, how do you get all rationals that are less than $rs$, if you have all rationals less than $r$ and all rationals less than $s$? Well, if $r$ and $s$ are both positive, you can look at all rationals $q_1$ and $q_2$ with $0\leq q_1\lt r$ and $0\leq q_2\lt s$, and consider $q_1q_2$. Every such product will be a positive rational less than $rs$; and every positive rational less than $rs$ will be such a product (though this is harder to establish). So we would like to define the product of positive Dedekind cuts that way: take all products $q_1q_2$ with $0\leq q_1\lt r$ and $0\leq q_2\lt s$; and then throw in all negative rationals to complete the Dedekind cut.
This is your definition here, but we need to check that it is indeed a Dedekind cut. breeden has given you good hints for doing this.
Now, what are examples of Dedekind cuts? Well, for any real number $a$, take $(-\infty,a)\cap\mathbb{Q}$. That's exactly what a Dedekind cut looks like (if you already know what the real numbers are).
If you don't know what the real numbers are, then for every rational $q$ the set
$$\{ r\in\mathbb{Q}\mid r\lt q\}$$
is a Dedekind cut. There are other cuts; the canonical example of a cut that is not of the form "all rationals less than the rational $q$" is:
$$\Bigl((-\infty,0)\cap \mathbb{Q}\Bigr)\cup \{r\in\mathbb{Q}\mid r\geq 0\text{ and }r^2\lt 2\}.$$