The simplest definition I know of symmetric set difference is
$$
x \in A \oplus B \;\equiv\; x \in A \not\equiv x \in B
$$
Using this, we can calculate the elements of the left hand side of first equality, as follows:
\begin{align}
& x \in A \oplus \varnothing \\
\equiv & \;\;\;\;\;\text{"definition of $\;\oplus\;$"} \\
& x \in A \not\equiv x \in \varnothing \\
\equiv & \;\;\;\;\;\text{"definition of $\;\varnothing\;$"} \\
& x \in A \not\equiv \text{false} \\
\equiv & \;\;\;\;\;\text{"logic: simplify"} \\
& x \in A \\
\end{align}
By set extensionality this proves the first equality.
To prove the second, we calculate similarly
\begin{align}
& x \in (A \oplus B) \oplus B \\
\equiv & \;\;\;\;\;\text{"definition of $\;\oplus\;$, twice; drop parentheses since $\;\not\equiv\;$ is associative"} \\
& x \in A \not\equiv x \in B \not\equiv x \in B \\
\equiv & \;\;\;\;\;\text{"logic: simplify"} \\
& x \in A \not\equiv \text{false} \\
\equiv & \;\;\;\;\;\text{"logic: simplify"} \\
& x \in A \\
\end{align}