The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful point of view that does not appear in loc. cit. is that this tensor product comes from the Lawvere theory of commutative monoids. To explain this, consider the $(2,1)$-category $\mathrm{Span}(\mathrm{Fin})$ whose objects are finite sets and whose morphisms are spans $I\leftarrow K\rightarrow J$. It has the following universal property: for any $\infty$-category $\mathcal C$ with finite products, there is an equivalence
$$
\mathrm{CMon}(\mathcal C) = \mathrm{Fun}^\times(\mathrm{Span}(\mathrm{Fin}),\mathcal C),
$$
where $\mathrm{Fun}^\times$ is the $\infty$-category of functors that preserve finite products. Since $\mathrm{Span}(\mathrm{Fin})$ is self-dual, this means that $E_\infty$-spaces are finite-product-preserving presheaves on $\mathrm{Span}(\mathrm{Fin})$:
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})).
$$
This was first studied in the thesis of J. Cranch. From this perspective, the direct sum and tensor product are the Day convolutions of $\sqcup$ and $\times$ on $\mathrm{Span}(\mathrm{Fin})$ (here $\times$ means the usual product of finite sets, which is not the categorical product in $\mathrm{Span}(\mathrm{Fin})$; the latter is the same as the categorical coproduct, i.e., the disjoint union $\sqcup$). For example, $E_\infty$-semirings can be described as right-lax symmetric monoidal functors $(\mathrm{Span}(\mathrm{Fin}),\times)\to(\mathcal S,\times)$ that preserve finite products.
The unit. As Rune already explained, the unit for the tensor product of $E_\infty$-spaces is the free $E_\infty$-space on a point, that is the groupoid $\mathrm{Fin}^\simeq$ of finite sets with the $E_\infty$-structure given by disjoint union. This is equivalently the presheaf on $\mathrm{Span}(\mathrm{Fin})$ represented by the point, which is the unit for $\times$ on $\mathrm{Span}(\mathrm{Fin})$.
Here are a few examples I could think of. Let $E\in \mathrm{CMon}(\mathcal S)$.
Tensoring with a free $E_\infty$-space. Let $X\in\mathcal S$. Then
$$
\left(\coprod_{n\geq 0} (X^n)_{h\Sigma_n}\right) \otimes E = \operatorname{colim}_X E,
$$
where the colimit is taken in $\mathrm{CMon}(\mathcal S)$. This follows from the case $X=*$ using that $\otimes$ preserves colimits in each variable.
Tensoring with $\mathbb S$. Tensoring with the sphere spectrum $\mathbb S$ is the same as group-completing:
$$
\mathbb S\otimes E = E^\mathrm{gp}.
$$
For example, for a ring $R$,
$$
\mathbb S\otimes \mathrm{Proj}(R) = K(R).
$$
where $\mathrm{Proj}(R)$ is the groupoid of finitely generated projective $R$-modules, and $K(R)$ is the K-theory space.
Tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$. Another localization of $\mathrm{CMon}(\mathcal S)$ is obtained by inverting integers (or rather, finite sets). The inclusion of the full subcategory of $E_\infty$-spaces on which multiplication by $n$ is invertible has a left adjoint $E\mapsto E[n^{-1}]$, which is equivalent to tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$. But unlike in the cases of either abelian monoids or spectra, $\mathrm{Fin}^\simeq[n^{-1}]$ is not just the sequential colimit of multiplication by $n$ maps; it is obtained from the latter by killing suitable perfect subgroups of its fundamental groups, in the sense of Quillen's plus construction, to ensure that $n$ acts invertibly.
Tensoring with $\mathbb N$. Let $\mathrm{FFree}_{\mathbb N}$ be the 1-category of finite free $\mathbb N$-modules. There is a functor
$$
\mathrm{Span}(\mathrm{Fin}) \to \mathrm{FFree}_{\mathbb N}
$$
sending a finite set $I$ to $\mathbb N^I$, inducing an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})) \stackrel{\mathrm{str}}\rightleftarrows \mathcal P_\Sigma(\mathrm{FFree}_{\mathbb N}).
$$
Objects in the right-hand side are sometimes called strictly commutative monoids (the group-complete ones are connective $H\mathbb Z$-module spectra). Tensoring with $\mathbb N$ amounts to strictifying a commutative monoid in this sense:
$$
\mathbb N\otimes E = E^\mathrm{str}.
$$
Unlike $\mathbb S$, $\mathbb N$ is not an idempotent semiring, that is, strictifying is not a localization. Indeed, $\mathbb N\otimes\mathbb N$ is an $E_\infty$-space whose group completion is the "integral dual Steenrod algebra".
Tensoring with $\mathrm{Vect}_\mathbb{C}^\simeq$. Let $\mathrm{Vect}_\mathbb{C}^\simeq=\coprod_{n\geq 0} BU(n)$, where $U(n)$ is regarded as an $\infty$-group (despite the notation, this is not really the core of an $\infty$-category of vector spaces). This is an $E_\infty$-space whose group completion is $\mathrm{ku}$. There is a related $\infty$-category $2\mathrm{Vect}_{\mathbb C}$ whose objects are finite sets and whose morphisms are matrices of complex vector spaces. As in the previous example we get an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})) \rightleftarrows \mathcal P_\Sigma(2\mathrm{Vect}_{\mathbb C}).
$$
An object in the right-hand side is roughly speaking a commutative monoid such that $U(n)$ acts on the multiplication by $n$ map in a coherent way. Tensoring with $\mathrm{Vect}_\mathbb{C}^\simeq$ gives the free commutative monoid with such structure.
