(Note: this isn't something I really know, so this might be wildly off base.)

To start with, let's look at a weaker transfer principle: the Jech-Sochor Embedding Theorem.

Jech-Sochor says that sentences depending only on a bounded amount of the cumulative hierarchy above the set $A$ of atoms can be "passed over" to a model of genuine ZF. More precisely, Jech-Sochor states that

Let $\gamma$ be a fixed ordinal, and $V$ a model of $ZFA$ with a set $A$ of atoms. Then there is a model $W$ of $ZF$ and an embedding $i: V\rightarrow W: x\mapsto \tilde{x}$ such that $(P_\alpha(A)^V, \in^V)\cong (P_\alpha(\tilde{A})^W, \in^W)$.

So statements of "fixed depth" can be transferred. For example, "non-well-orderability" can be preserved by setting $\gamma=\omega+2$ (really, we just care about the powerset, but we also want to talk about maps from $\omega$ so we need to go up $\omega+1$ many levels to get $\omega$ into $P_\gamma(A)$, and then one more level to get the desired maps).

In Pincus, this property of the truth of $\phi(X)$ only depending on some fixed level of the cumulative hierarchy over $X$ is called *boundability*; so, for example, on page 722 Pincus phrases the Jech-Sochor theorem as:

"A boundable statement is transferable."

The question is whether we can improve this result to transfer statements that don't necessarily depend just on $P_\gamma(X)$ for some fixed $\gamma$, but are still "locally determined" in some sense. This "locally determined" is his condition that

$\vert x\vert\le\sigma(y)$

in a (sur/in)jectively boundable statement. So now we've switched from caring about the number of powersets required to reach a set, to caring about its "cardinality" being small. Note, though, that the bound on the size of $x$ itself must be boundable, so the idea of Jech-Sochor isn't really going away.

The requirement that each element of $x$ have no intersection with the transitive closure of $y$ seems more technical, and I'm not sure if there's a clean intuition behind it. His example 2B1 shows why I feel okay not caring about this part of the definition too much - in the end, we take some class of potential counterexamples (field expansions that *might* be algebraic closures) but which don't satisfy this disjointness condition, and just slide them over in an appropriately definable manner (in this case, $x\mapsto \lbrace (w, y): w\in x\rbrace$). I suspect that in general something like this will be possible without much difficulty (although I am not sure on this point).

So what we're left with is that a (sur/in)jectively bounded sentence is essentially a $\Sigma_2$-sentence where

the universal quantification is taken over sets of boundable size, and

the matrix of the sentence is boundable in the original Jech-Sochor sense.

(Of course, this isn't really $\Sigma_2$, since this "matrix" might well have quantifiers, but oh well.) What's really new here is this universal quantifier - note that

At this point it would be nice to see a injectively boundable statement which is not boundable. I think the clearest example is Pincus' 2B6 on page 724 (actually, he uses this as an example of an injectively boundable statement which is not surjectively boundable, but that distinction seems less intuitively crucial to me). The statement here is

"Every infinite partially ordered set has either an infinite chain or an infinite antichain but there is an infinite, Dedekind-finite set."

This sentence is a conjunction $\Phi\wedge\Psi$, where

$\Phi\equiv$ "Every infinite poset has a chain or antichain," and

$\Psi\equiv$"There is a strictly Dedekind-finite set."

Now $\Psi$ already transferable by Jech-Sochor (since "is a Dedekind-finite set" depends just on the powerset). $\Phi$, however, is a bit trickier, since it involves quantifying over the class of all posets! And this certainly can't be done with Jech-Sochor.

Instead, we use a trick. First, we can rewrite $\Phi$: $$\Phi\equiv "\forall x(\vert x\vert_-\le\omega\implies (\text{ if $x$ is an infinite poset, then $x$ has an infinite chain or antichain})) "$$ since if $\vert x\vert_->\omega$ then we can already build a chain or antichain without choice. Now the conclusion of this implication is a boundable formula of $x$, since it really only talks about the powerset! So even though the whole sentence $\Phi$ wasn't boundable, by massaging it a bit we got it to the point where the universal quantifier causing all the trouble was just over sets of small "size," and this was enough for it to be injectively boundable.

Note that here, the specific notion of size we use is crucial: $\Phi$ isn't surjectively boundable, since $\vert x\vert^-$ can behave more weirdly on Dedekind-finite sets. So this is what should motivate injective boundability: it's the broadest obvious way to push up the strength of Jech-Sochor to allow some non-powerset-bounded universal quantification.

Hopefully, this helps. Tl;dr: injectively-bounded properties will generally look like "This simple property $(\neg\Psi)$ does not always $(y)$ have small witnesses $(\forall x[\vert x\vert_-< . . .])$."