It's quite simple: in a structure with a single object - and for simplicity, let's assume only unary relations at first - we think of each unary relation as a single atomic proposition, and of the single object as a "propositional world." (What about multiple worlds? Well, this takes us into propositional modal logic, and explains why that can be thought of as a fragment of first-order logic.) Quantifiers are stripped away, since they only point to the single object, and the unary predicates get replaced by corresponding propositional atoms. For an example of how this works, the first-order sentence $$\exists x\forall y(P(x)\vee Q(y))$$ is thought of as the propositional sentence $$p\vee q.$$
Binary relations are no different from unary relations, since there is only one object; so similarly, $$\exists x\forall y(P(x)\vee (Q(y)\wedge R(x,y)))$$ would be thought of as $$p\vee (q\wedge r).$$ And function symbols can be replaced by relations naming their graphs.
Specifically, here's the theorem:
Let $\varphi\mapsto\varphi^*$ be the map from first-order sentences to propositional sentences described (informally) above. Then for every first-order sentence $\Sigma$, we have $$\Sigma\cup\{\forall x,y(x=y)\}\vdash_{first-order} \varphi\quad\iff\quad \Sigma^*\vdash_{propositional} \varphi^*$$ (where $\Sigma^*$ is shorthand for $\{\sigma^*:\sigma\in\Sigma\}$).