\begin{align*} P(B \cup C | A) & = \frac{P(A \cap (B \cup C)}{P(A)} \\ & = \frac{P( (A \cap B) \cup (A \cap C) )}{P(A)}\\ & = \frac{P(A \cap B)}{P(A)} + \frac{P(A \cap C)}{P(A)}\\ & = P(B | A) + P(C | A). \end{align*}