There is little Shannon or information theory in the quantity you are examining, which is just the 19th century definition of classical entropy of probability distributions for $f(x)=|\psi(x)|^2$. There is also little actual quantum mechanics, as all you need is the above definition of your probability distribution, without regard of where it came from, or reference to the interference quantum effects associated with ψ, etc.
As Clausius' name implies, entropy describes probability dispersal, disorder, or randomness (τροπή ~ change).
Nevertheless, information scientists have adopted/appropriated it as
differential entropy for lack of information (randomness); and there is no point in splitting hairs about names; in QM uncertainty discussions the name has stuck. At the end of the day, the above-defined classical entropy S does help with QM quite a bit, after all.
A normalized probability distribution $f(x)$ may be broad or narrow, which is what the standard deviation σ (width), or variance (its square), describes. First look at a Gaussian/normal distribution,
$$
f(x)= \frac{1}{\sigma\sqrt{2\pi}} e^{-(x/\sigma)^2/2} \qquad \leadsto \\
S= \ln ~(\sigma \sqrt{2\pi e}),
$$
so S increases with the variance.
Small variance indicates localization, and the limit of $\sigma\to 0$ is a δ-function spike: ultra-localization; meaning no uncertainty in the position of your wave packet (but an infinite uncertainty in its corresponding momentum). (It's up to you how to interpret this infinite information/negentropy, if you are hung up on information theory.)
Conversely, huge σ (huge S) signals huge dispersal of the probability distribution, on its way to terminal delocalization. So far, so good.
But... You may easily imagine a probability distribution consisting of two Gaussians with their centers at some distance a from each other and small widths. σ will be more like this distance (ultimately a/2), rather than the smaller widths of each constituent Gaussian. (Proceed to calculate such examples.) You know this distribution is less ordinary than an unremarkable Gaussian, and has more information content, (and is more localized in space, since the probability vanishes everywhere except in the two narrow bumps a apart).
You'd like to have some measure of this different than σ ~ a/2, and to indicate your two-bump distribution is more localized than a single Gaussian with the same σ. Hopefully, you found that this S is less than the S of that single Gaussian: S is a better measure of delocalization than σ. In fact, there is a theorem.
You know random things end up in Gaussians, and for a given σ, the Gaussian is the maximum entropy distribution; the most delocalized distribution for a given σ. The normal distribution is where information goes to die.
You may have fun experimenting with comb distributions, whose teeth are Gaussians, of the same σ, and thus further lower S; this is even more extraordinary and ordered/non-random: more informative. Try the ones listed.
The takeaway is that S is a far better measure of fine detail and localization than the crude variance, and, in fact, as a result, it provides a more stringent uncertainty inequality for the uncertainty principle, entropic uncertainty, linked above.
The actual entropy of a quantum state, however, is not what you are discussing. The quantum information entropy for QM is the von Neumann entropy, whose definition in phase-space outranges this discussion. As mentioned in a comment by @Lucas, for a pure state, the von Neumann entropy actually vanishes: you have maximum information about that state, and you are in no doubt about it — except some observables which are mutually incompatible and simultaneously unknowable. But that is another, more exciting story.
