What does $\forall a \in A \exists b \in B(b \in C \rightarrow a \in C)$ mean? What does $\forall a \in A \exists b \in B(b \in C \rightarrow a \in C)$ mean?
I know it means that for all $a$ in $A$ there exists a $b$ in $B$ such that $b$ in $C$ implies $a$ in $C$, but what does that mean intuitively, because to me the interaction between the $\exists$ and the implication doesn't make any sense.  It seems to be saying that there exists a $b$ in $B$ and if that $b$ happens to be in $C$, then it implies that $A \subseteq C$?  Does it means that the entire set of $A$ is that one element of $B$ and it's also in $C$?

It is probably easiest to understand this by using the rule that $P\to Q$ means the same as $\neg P\lor Q$. So what we have here is
$$ \forall a\in A. \exists b \in B. (b\notin C \lor a \in C)$$
Now, let's consider the subformula $\exists b \in B. (b\notin C \lor a \in C)$. It should be clear that if there is a $b$ such that $b\notin C$, then this is true no matter what $a$ is. On the other hand, if $a\in C$, then it is true if only $B$ contains some $b$. So this is equivalent to $(\exists b\in B. b\notin C) \lor (B\neq \varnothing \land a \in C)$. So we have
$\forall a \in A. \bigl( (\exists b\in B. b\notin C) \lor (B\neq \varnothing \land a \in C)\bigr)$
Now again this is certainly true if $\exists b\in B. b\notin C$ is. Otherwise we must have $B\neq\varnothing$ and $a\in C$ for all $a\in A$. So it is equivalent to
$$(\exists b\in B. b\notin C) \lor (B \neq \varnothing \land \forall a\in A. a\in C)$$
We can rewrite that to use set algebra instead of logic, so see that what the original formula "really" means is
$$ B\setminus C \neq \varnothing \lor (B \neq \varnothing \land A\subseteq C)$$
