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 $C$?

Note that $b\in C\to a\in C$ is equivalent to $b\notin C\lor a\in C$. Thus, it says that there is a function $f:A\to B$ with the following property:

for each $a\in A$, either $f(a)\notin C$, or $a\in C$.

If $A\subseteq C$, this is guaranteed to be the case provided that $B\ne\varnothing$: we can pick any $b_0\in B$ and let $f(a)=b_0$ for each $a\in A$. 
If $A\nsubseteq C$, fix $a_0\in A\setminus C$. In order for the highlighted statement to be true, we must be able to define $f(a_0)\in B$ in such a way that $f(a_0)\notin C$, which means that $B\setminus C$ must be non-empty. Conversely, if there is a $b_0\in B\setminus C$, we may let $f(a)=b_0$ for all $a\in A$, and this $f$ will have the desired property.
All of this boils down to saying that the statement is true if and only if $B\ne\varnothing$, and either $A\subseteq C$ or $B\nsubseteq C$.
