I used the logic in the title in this answer. Is this implication true and is there a simple explanation of why it is true?
No, this is not a valid implication.
$(A\land B)\Rightarrow (A\lor C)$ is a tautology, and $A\Rightarrow B$ is not. So the former cannot imply the latter.
For example, if $A$ is true but $B$ and $C$ are false, then $(A\land B)\Rightarrow (A\lor C)$ is true, but $A\Rightarrow B$ is false.