One way to think of these spectra is in terms of the cohomology theories they define. In other words, if
$E$ is a spectrum, what is $[E, I\mathbb Z]$? This is less of a description of what they are and more of a
description of what they do; there are probably other, more conceptual answers to your question.
1. $I\mathbb Z$ and $\Sigma^n I\mathbb Z$
The universal coefficient theorem describes how to compute cohomology groups from homology groups: there is a short
exact sequence
$$ 0\longrightarrow \mathrm{Ext}^1(H_{n-1}(X), \mathbb Z)\longrightarrow H^n(X)\longrightarrow \mathrm{Hom}(H_n(X), \mathbb
Z)\longrightarrow 0,$$
and it splits noncanonically.
If you try to do this for generalized
cohomology, nothing so nice
is true, and the whole story is more complicated.
Nonetheless, part of the story can be salvaged: if you try this with stable homotopy groups (the homology theory
represented by the sphere spectrum), you obtain the cohomology theory represented by the Anderson dual of the
sphere. That is, for any spectrum $X$ there is a short exact sequence
$$ 0\longrightarrow \mathrm{Ext}^1(\pi_{n-1}(X), \mathbb Z)\longrightarrow [X, \Sigma^n I\mathbb Z]\longrightarrow
\mathrm{Hom}(\pi_n(X), \mathbb Z)\longrightarrow 0,$$
and it splits noncanonically. (See Freed-Hopkins, §5.3.)
There's a more general version of this in skd's answer.
2. $I\mathbb C^\times$ and $\Sigma^n I\mathbb C^\times$
These spectra provide an analogue of Pontrjagin duality. If $A$ is an abelian group, the set of maps $A\to\mathbb
C^\times$ is an abelian group under pointwise multiplication, and this is called the Pontrjagin dual of $A$. An
analogue for spectra might be the assignment $X\mapsto \mathrm{Hom}(\pi_nX, \mathbb C^\times)$, the group of
characters of the $n$th homotopy group of $X$. This is precisely what $[\Sigma^n X, I\mathbb C^\times]$
is; more broadly, one could describe $I\mathbb C^\times$ as the spectrum whose cohomology theory is calculated by
$$(I\mathbb C^\times)^n(X) = \mathrm{Hom}(\pi_{-n}(X), \mathbb C^\times).$$
As an addendum, I'm guessing this question arose because of the appearance of these spectra in physics,
specifically in the classification of invertible topological field theories. Following Freed-Hopkins, the
classification of invertible TQFTs $\mathsf{Bord}_n\to \mathsf C$, where $\mathsf C$ is some target symmetric
monoidal $(\infty, n)$-category, is equivalent to the abelian group of homotopy classes of maps between the
classifying spectrum of the groupoid completion of $\mathsf{Bord}_n$ and the classifying spectrum of the groupoid
of units of $\mathsf C$. The classifying spectrum of $\mathsf{Bord}_n$ is determined by
Schommer-Pries, and for certain reasonable choices of $\mathsf C$, we get
$\Sigma^nI\mathbb C^\times$ and $\Sigma^{n+1} I\mathbb Z$ (for small $n$, and conjecturally for all $n$).
For small $n$, we know some good chocies for $\mathsf C$: for example, if we let $n = 1$, we can take $\mathsf C$
to be the category of super-vector spaces, and for $n = 2$ we can take the Morita 2-category of superalgebras. In
both cases, the classifying spectrum is the connective cover of $\Sigma^n I\mathbb C^\times$. This suggests that in
higher dimensions $n$, we might find symmetric monoidal $n$-categories $\mathsf C$ whose classifying spectra
continue this pattern, seeing more and more of $I\mathbb C^\times$, and the calculations of SPT phases coming out
of physics provide heuristic evidence for this conjecture.
This used the discrete topology on $\mathbb C$. If you instead give $\mathbb C$ the usual topology when defining the
categories of super-vector spaces or superalgebras, you get different classifying spectra: the connective covers of $\Sigma^{n+1} I\mathbb Z$ for $n =
1$, resp. $2$. Conjecturally, this pattern also continues further. Using the usual topology on $\mathbb C$ corresponds to classifying
deformation classes of invertible TQFTs rather than isomorphism, and again physics calculations provide some evidence for the conjecture.
