I would also like to know the answer to your question. Since no one has given an answer yet, I'll speculate recklessly and irresponsibly on how this might work (by riffing off of the final paragraph of section 5.1 of the survey).

Before figuring out what "higher equivariant homotopy theory" is, we should first know what "equivariant homotopy theory" is. In this case, we want "global equivariant homotopy theory", aka the homotopy theory of smooth stacks, as introduced by Gepner-Henriques.

They show that the homotopy theory of smooth stacks is modelled by the homotopy theory $PSh(Orb)$ of presheaves of infinity-groupoids on $Orb$. Here $Orb$ is a topologically enriched category (i.e., infinity category) whose objects are compact Lie groups $G$, and whose morphism spaces are given by
$$Orb(G,H) := BFun(G,H),$$
a classifying space of the category of smooth functors (so path components of $Orb(G,H)$ correspond to conjugacy classes of homomorphisms $\phi\colon G\to H$, and the path component containing $\phi$ is the classifying space of the centralizer of $\phi$)

Remark: there is an evident map $B\colon Orb(G,H)\to Map(BG,BH)$. When $H$ is an extension of an abelian group by a torus, it is a weak equivalence; but it is not generally a weak equivalence otherwise.

Jacob's construction apparently associates to each derived oriented elliptic curve $E\to Spec(A)$ a functor $M\colon Orb\to Sch_A$ taking values in derived schemes over $A$. The curve $E=M(U(1))$. Section 5.1 suggests extending this construction by incorporating a "level".

So let's define $\widetilde{Orb}$ to have objects $(G,\ell)$, where $G$ is a compact Lie group and $\ell\colon BG \to K(\Lambda, 4)$ a map, where $\Lambda$ is a free abelian group. The morphism space $\widetilde{Orb}((G,\ell), (G',\ell'))$ is the homotopy pullback of
$$Orb(G,G') \to Map(BG,BG') \to Map(BG, K(\Lambda',4)) \leftarrow Map(K(\Lambda,4), K(\Lambda',4)).$$
I.e., a "map" $(G,\ell)\to (G',\ell')$ is a homomorphism $\phi\colon G\to G'$ together with $f\colon K(\Lambda,4)\to K(\Lambda',4)$ and a homotopy $f\circ \ell \sim \ell'\circ B\phi$.

Now we can *define* 2-equivariant homotopy theory to be $PSh(\widetilde{Orb})$

Ideally, at this point you would identify $\widetilde{Orb}$ with something more geometric, e.g., some suitable category of Lie 2-groups. Maybe you would even identify $PSh(\widetilde{Orb})$ with the homotopy theory of mumble mumble smooth 2-stacks mumble (I have no actual idea here). However, we don't need to do this if we just want to play with homotopy theory: we just need $\widetilde{Orb}$.

Note that $Orb$ can be identified as the full subcategory of $\widetilde{Orb}$ consisting of $(G,0\colon BG\to K(0,4))$. I think Jacob is claiming that the functor $M\colon Orb\to Sch_A$ can be extended to a functor from $\widetilde{Orb}$. This wouldn't be an arbitrary extension, but would satisfy some compatiblities, perhaps including:

Compatibility with base change: given a homomorphism $\phi\colon H\to G$, require that $M(H,\ell\circ B\phi)\to M(G,\ell)$ be the evident base change along $M(H,0)\to M(G,0)$.

Given two levels $\ell\colon G\to K(\Lambda,4)$ and $\ell'\colon G\to K(\Lambda',4)$, $M(G,\ell\oplus \ell')$ should be the pullback of $M(G,\ell)\to M(G,0)\leftarrow M(G,\ell')$. This would imply that that the values of $M$ are determined by those on objects on $Orb$, together with $(G,\ell)$ with $\ell\colon BG\to K(Z,4)$.

The previous implies that $M(e,Be\to K(Z,4))$ has the structure of an abelian group object in $Sch_A$. We would require that this object be $\mathbb{G}_m$.

Require every $M(G,BG\to K(Z,4))$ be a $\mathbb{G}_m$-torsor.

Then Jacob's theorem is possibly something like: an extension of $M$ satisfying a list of compatibilities is entirely determined by specifying a value at the cup-product level $\ell\colon BU(1)\times BU(1)\to K(Z,4)$, which itself should satisfy some properties/structure ("symmetric biextension of $M(U(1))=E$ by $\mathbb{G}_m$").

One might guess that n-equivariant homotopy theory is built from pairs $(G,\ell)$, where $\ell \colon BG\to Z$ is a map to some suitable type of $(n+2)$-truncated space $Z$. How this ought to be specified probably depends on what you want to accomplish with it. I don't have any idea about that.