Suppose you have a dynamic system
$$ x_{t+1} = Ax_{t} $$
with a stationary point (or steady state as used in growth or RBC literature), say, $x^*$, i.e. $x^{*} = Ax^{*}$.
Now, consider the following question. Starting from an initial value $x_0$, how many paths are there leading to the stationary point $x^*$? If there is an unique path going from $x_{0}$ to $x^{*}$, then your model is well behaved in the sense that you can track the vector of variables, $x_t$, along the transition without worrying which path you are actually on. This is the saddle-path stability case that you are referring to. On the other hand, if the answer is not affirmative, it means that you have at least two routes from $x_0$ to $x^{*}$. There is another case: no matter where you start from you eventually end up at the stationary point $x^{*}$, in which case your model is said to be indeterminate.
So you can think of saddle-path stability as a desirable feature that you would like that your model to manifest in order to analyze the problem at hand. For example, standard RBC models all posses this property.
There are some mathematical conditions that ensure the saddle-path stability.
For the details and more, check out the section 7.8 (The q-Theory Investment and Saddle-Path Stability) in Introduction to Modern Economic Growth (Acemoglu, 2009).
Ok, I'll try to explain intuitively for the Ramsey model (but it's not rigorous). Suppose that from the first order conditions, you can conclude that the equilibrium is unique and furthermore you can express consumption as a function of capital stock, that is $c_{t} = g(k_t)$, for some (smooth) function $g$. If that is the case, for a given initial capital stock $k_0$, you know how the economy is going to evolve. That is, if we denote the vector of variables by $x_t = (c_t, k_t)$, you know $\{ x_{t}\}_{t=0}^{\infty}$; because
$t=0: \quad$ $k_0$ is given, and $c_0 = g(k_0)$,
$t=1: \quad$ $k_1 = (1-\delta)k_0 + f(k_0) - c_0$, and $c_1 = g(k_1)$,
(here $f$ is the production function and $\delta$ is depreciation)
$\vdots $
$t=\tau: \quad$ $k_{\tau} = (1-\delta)k_{\tau -1} + f(k_{\tau -1}) - c_{\tau -1}$, and $c_{\tau} = g(k_{\tau})$,
so on ....
Now, suppose the economy loses some of its capital stock for some reason while it was on $x^*=(k^*, c^*)$ and the new capital stock is, say, $k_{\tau}$.
Where will the economy be with this new capital stock?
Well, we have just claimed that there is a corresponding consumption $c_{\tau}=g(k_{\tau})$, and therefore the economy will jump to the point $x_{\tau}=(k_{\tau}, c_{\tau})$ on impact the shock and then move toward back to steady state $x^*$. You can think of $x_{\tau}$ as the red point in the figure.
Finally, a few remarks on the intuition behind the mathematical conditions ensuring saddle-path stability.
1) Note that, in the model we have actually two variables $c_t$ and $k_t$, but we have assumed that one of them can be expressed in terms of the other, so the number of free variables is reduced to one, which is $k_t$ here. Basically, in order to be able to do this, you want to have a smooth curve mapping capital stock to consumption, i.e. $g: \mathcal{R} \to \mathcal{R}$ is a smooth curve (1-dimensional manifold).
2) Once the economy is at $x_{1}$ after the shock, you want it go back to $x^*$ and this is related to the number of eigenvalues that are less than one in absolute value.
For precise statements (in a general environment) see Acemoglu as I mentioned before.
Regarding rational expectations(RE): it is a solution concept and RE alone does not imply saddle-path stability.
