ORIGINAL RESPONSE:
There is a lot more to this definition that I haven't talked about, so this is a very rough summary. I urge you to take a look at the book instead of just taking my word (I am in no ways an expert).
Addendum:
I've recently circled back to this problem after nearly 5 years, and I now realize that the actual argument for this proof isn't as complicated as I thought it was.
For anybody reading this, here is a simple definition of a logic, so that VP is equivalent to the statement that every logic has a strong compactness cardinal.
A logic consists of two parts:
- The language, which is a proper class $\mathcal{L}$ of $\tau$-sentences for each relational signature $\tau$; these are functions $\varphi:x\rightarrow\tau$ for some set $x$. The class of $\tau$-sentences for a particular $\tau$ may be denoted $\mathcal{L}[\tau]$.
- The semantics, which is a proper class $\models_{\mathcal{L}}$ of tuples $(\mathcal{M},\varphi)$, where $\mathcal{M}$ is a $\tau$-structure for some signature $\tau$ and $\varphi\in\mathcal{L}$ is a $\tau$-sentence.
We require that it satisfies the following conditions:
- Isomorphism invariance: if $\mathcal{M}$ is isomorphic to $\mathcal{N}$, then for any $\varphi$ in $\mathcal{L}$, $$(\mathcal{M}\models_{\mathcal{L}}\varphi)\leftrightarrow(\mathcal{N}\models_{\mathcal{L}}\varphi)$$
- Reduction invariance: if $i:\tau\rightarrow\sigma$ is a monomorphism of relational structures, and $\mathcal{M}$ is a $\sigma$-structure, then for any $\varphi:x\rightarrow\tau$ in $\mathcal{L}$, the composition $i\circ\varphi:x\rightarrow\sigma$ is also in $\mathcal{L}$, and $$(\mathcal{M}|_{\tau}\models_{\mathcal{L}})\varphi\leftrightarrow(\mathcal{M}\models_{\mathcal{L}} i\circ\varphi)$$
These properties essentially guarantee that each $\varphi$ is a statement, with parameters in $V$, about the relations in the structure; and nothing more.
Of course, in the above formulation, logics like $\mathcal{L}_{\infty,\infty}$ can be construed, which have no strong compactness cardinal no matter how you slice it. We can say that the logic is small when there is some fixed set $X$ such that each $\tau$-sentence $\varphi\in\mathcal{L}$ is a function $x\rightarrow\tau$ for some $x\in X$.
As an example, $\tau$-sentences of $\mathcal{L}_{\kappa,\kappa}$ may be construed as functions $x\rightarrow\tau$ for $x\in H_\kappa$. So, this logic is small.
The proof:
VP is equivalent to the statement that for every class $A$ there is a stationary class of $A$-extendible cardinals. So, assuming VP, there is a $\models_{\mathcal{L}}$-extendible cardinal $\kappa$ such that the set $X$ mentioned above is in $V_\kappa$.
Let $\Sigma\subset\mathcal{L}[\tau]$ be such that for every $t\subset\Sigma$ with $|t|<\kappa$, there is a $\tau$-structure $\mathcal{M}_t$ such that $\mathcal{M}_t\models_{\mathcal{L}}\varphi$ for all $\varphi\in t$. In other words, let $\Sigma$ be a $\kappa$-satisfiable $\mathcal{L}$-theory. Then, let $j:(V_\eta;\in,\models_{\mathcal{L}})\rightarrow (V_\theta;\in\models_{\mathcal{L}})$ be an elementary embedding with critical point $\kappa$, such that $\Sigma\in V_\eta$ and each $\mathcal{M}_t\in V_\eta$.
Since $V_\eta$ witnesses that every subset of $\Sigma$ of size below $\kappa$ has a model, by elementarity and $\models_{\mathcal{L}}$-correctness, $V_\theta$ witnesses that every subset of $j(\Sigma)$ of size below $j(\kappa)$ has a model. In particular, $j"\Sigma$ is a subset of $j(\Sigma)$ of size $\kappa$, so it must have a model $\mathcal{M}$. However, we are not quite done.
This $\mathcal{M}$ is a $j(\tau)$-structure, satisfying every member of $j"\Sigma$. So, we don't quite have a model of $\Sigma$ per se. However, every $\varphi\in j"\Sigma$ is of the form $j(\varphi)$ for $\varphi:x\rightarrow\tau$. *Because $x\in V_\kappa$, we have that $j(\varphi)=j|_\tau\circ\varphi:x\rightarrow j(\tau)$, and $j|_\tau:\tau\rightarrow j(\tau)$ is a monomorphism of signatures.
So, by reduction invariance, $\mathcal{M}|_\tau$ is a model of every $\varphi\in\Sigma$, completing the proof.
The reverse direction is not as hard. Given a proper class $A$ of $\tau$-structures, we wish to find a first-order elementary embedding between two of its members. Because first-order logic is isomorphism invariant, WLOG $A$ is closed under isomorphism. For each monomorphism $i:\tau\rightarrow\sigma$, we can consider the class $A^i$ of $\sigma$-structures who reduce to a member of $A$. If there is an elementary embedding in $A^i$ for some $i$, then since first-order logic is reduction invariant, there is an elementary embedding in $A$.
So, we construct the logic by starting with first-order logic, adding a $\tau$-sentence satisfied only by members of $A$, and then for each monomorphism $i:\tau\rightarrow\sigma$ adding a $\sigma$-sentence satisfied only by members of $A^i$. This logic is readily seen to be isomorphism invariant, reduction invariant, and small by the exposition above. Finally, if it has a strong compactness cardinal, pick any structure $\mathcal{M}$ in $A$ of size greater than that of the cardinal, and consider its elementary diagram in this logic. Add to it a new constant symbol and, for each $x\in\mathcal{M}$, an axiom stating that $x$ is not equal to the new constant symbol. This theory is clearly $\kappa$-satisfiable, so by strong compactness it is satisfiable. But any model of it must be a member of $A$ admitting a nontrivial elementary embedding from $\mathcal{M}$. QED
