One clue that might lead you to Dedekind cuts is as follows.
Suppose you are oblivious to the real numbers, and only know about the rationals. One day in physics class you're studying the path of a projectile under constant acceleration. In an example you come up with the equation $y = 2 - x^2$ where and $y$ is height and $x$ is time. Solving for height equal to zero you get $x^2 = 2$ or more specifically $2{q^2} = p^2$ if you express $x$ as $\frac{p}{q}$ (where $p$ and $q$ are positive integers).
But then comes the problem. That last equation has no solution. The proof isn't automatic, but essentially $2{p^2}$ has an odd number of $2$'s in it's factorization, and $q$ must have an even number. This is because of what happens if you square the factorization of an integer... the powers afterwards must all be even, right? So it's established that this equation (and many like it) have no solution.
Now if you dig a pit and repeat the experiment so the projectile can pass below height zero you find a very odd situation. A physical object is above a certain plane and then after a period of time is below a certain plane. In the course of it's travel, did it just skip through that plane? Physical intuition suggests that it could not... even in a strictly geometric sense it's hard to imagine a point just phasing through a plane like that. So perhaps the rationals need to be extended to these new 'real' quantities just like the integers needed to be extended to the rationals.
With this in mind, you have a dilemma: how does one approach these new 'real' quantities if they haven't yet been defined. One idea is to use elimination. Whatever this new quantity is (soon $\sqrt{2}$) in comparison to the rationals it is greater than any rational $r$ with $r^2 < 2$ and less than any rational $s$ with $s^2 > 2$. In the experiment, these are the (rational) times before and after the point passed through the plane. This gives you two intervals of rational numbers... an interval being defined by the property that if it contains two numbers, it contains any number in between those two.
Now, to repeat, we can't solve for the time of intersection using the rationals but we can pick two rational times just before and just after our 'missing' intersection. With some work, we can even pick these two as close apart as we want. This is a good clue that as far as approximation goes, the two intervals are doing a good job. We might also notice that we can just keep track of one of the intervals, say the first one, and we'll remember the second for free: it's just the set of rationals greater than all the rationals in the first set.
After more examples like this, we would see more and more examples of intervals of rational numbers that are unbounded below and bounded above. Some are new, some we have already seen before all this speculation, like $(-\infty,r)$. This is convenient, as it provides a way to represent the rationals using these new intervals.
The last step is to somehow define the various mathematical operations for these placeholder intervals so that we can work with the new quantities by using their known properties, and not by how we represent them using Dedekind cuts. This is fairly obvious for Dedekind cuts that represent positive quantities if you remember to first remove the negative, then do the operation on the sets involved, and then add the negatives back in.
[Note: My explanation is all very loose and rough. Also, it turns out that adding roots of polynomials whenever they cross the $x$-axis is not enough. The real numbers add these, and more, so that there is also always a solution to $f(x) = 0$ whenever you have $f(-1) > 0$ and $f(1) < 0$ for a decreasing function $f$ defined on $[-1,1]$, not just polynomials, etc...]