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}