What does $P(x) : x = -3$ mean in this question?
Consider the open sentences
$$ P_1(x): x = -3. \text{ and } P_2(x): |x| = 3, $$
where $x \in \mathbb{R}$, that is, where the domain of $x$ is $\mathbb{R}$ in each case. We can then form the following open sentences:
\begin{align} \unicode{8764} P_1(x) : x &\ne -3. \\ P_1(x) \lor P_2(x): x &= -3 \text{ or } |x| = 3. \end{align}
The author apparently uses the symbol $:$ to say what a purely symbolic expression means.
That is, $P_1(x) : x = -3$ denotes the fact that $P_1(x)$ means '$x = -3$'; as such, $P_1(x)$ is defined to be the sentence '$x = -3$'. Likewise, $P_2(x)$ is defined to be the sentence '$|x|=3$'.
Then ${\sim} P_1(x) : x \ne -3$ denotes the fact that the symbolic expression '${\sim} P_1(x)$' means that $x \ne -3$. And so on.