What is meant by $P(X = x)$? What is meant by the statement $P(X = x) = \theta$? As in, what is its English translation? I'm assuming that $X$ is a random variable and $x$ is a member of its sample space. 

Is it just "the probability that event $x$ occurs for random variable $X$ is equal to $\theta$"?

If so, why use the "$=$" sign inside the parentheses? Isn't that technically misleading since the random variable $X$ is not equal to one of its sample space members $x$?
The statement is "The probability that the random variable $X$ takes the value $x$ is equal to $\theta$." The notation is a bit misleading. You are right that $X$ does not equal $x$, but rather, after a measurement of $X$ is made, the value $x$ is observed as a result.
The statement $\Bbb P(X=x)=\theta$ reads 


  The probability that $X$ is equal to $x$ is $\theta$.


For example, consider the experiment of rolling one standard fair six-sided die once. Then our sample space is
$$
S=\{1,2,3,4,5,6\}
$$
Now, let $X:S\to\Bbb R$ be given by the formula
$$
X(x)=
\begin{cases}
1 &amp; x\text{ not prime} \\
2 &amp; x \text{ even prime} \\
3 &amp; x\text{ odd prime} 
\end{cases}
$$
Then
\begin{align*}
\Bbb P(X=1)&amp;=\frac{1}{2}&amp;\Bbb P(X=2)&amp;=\frac{1}{6}&amp;\Bbb P(X=3)&amp;=\frac{1}{3}
\end{align*}

Note that the restrictions on the possible values of $\theta$ are quite severe if our random variable is continuous. Indeed, let $X$ be a random variable with probability density function $f$. Then
$$
\Bbb P(X=x)=\Bbb P(x\leq X\leq x)=\int_x^x f(t)\,dt=0
$$