The teach us think "if conditions hold" then "Are these statements true" but I find it better and more into intuitive to think if these statements are true, what is possible.
If $A \implies B$ what is possible?
Is $A$ possible? yes but only if $B$ is; If $A$ then $B$.
Is $A$ being false possible? Sure, $A\implies B$ doesn't say anything about what happens if $A$ is false.
Is $B$ possible? Sure, if $A$ then $B$ must be true. And if not $A$ then anything is possible.
Is $B$ false possible? Well, not if $A$ is true it's not but if $A$ is false that's just fine.
What about $A$ and $B$ etc. Well, $A$ and $B$ and $A$ and not $B$ and not $A$ and not $B$ are all possible but only $A$ and not $B$ are impossible. That can't ever happen (if we accept $A \implies B$) so one way or another we are either going to not have $A$ or we will have $B$.
Now do the same for $\lnot A \lor B$. You get the same results. That two statements can only be true under the same conditions.