How to interpret material conditional and explain it to freshmen? After studying mathematics for some time, I am still confused.

The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the symbol “$\Rightarrow$”, which is read as “implies” or “if … then ….” It is customary and reasonable to treat “$\Rightarrow$” as the material conditional, i. e. as equivalent to saying “the antecedent is false or the consequence is true (or both)”.

Now there is some controversy about whether the material conditional really captures conditional statements because it doesn't really say anything about a causal connection between the antecedent and the consequence. This is quite often illustrated by the means of statements in natural languages such as “the moon is made out of cheese $\Rightarrow$ all hamsters are green” – since the moon isn't made out of cheese, is this statement true? This remained problematic to me.

While I came to accept the material conditional as a good way of describing implications and conditionals, I'm having a hard time to explain this usage to freshmen whenever I get asked.

My questions are: How can we best justify the interpretation of “$\Rightarrow$” as a material conditional? Why is it so well-suited for mathematics? How can we interpret or read it to understand it better? Can my confusion about it be led back to some kind of misunderstandig or misinterpretation of something?

I have yet very poor background in mathematical logic (I sometimes browse wikipedia articles about it), but I'd have no problem with a technical answer to this question if it clarifies the situation.
Perhaps this will help to capture the truth-functional character of material implication:

The truth-value of an inclusion (subset) relation between sets corresponds to the truth-value of an implication relation, where $\subseteq$ corresponds to the $\rightarrow$ relation. 

E.g., suppose $A\subseteq B$. Then if it is true that $x\in A$, then it must be true that $x\in B$, since $B$ contains $A$. However, if $x\notin A$ (if it is false that $x \in A$), it does not mean that then $x\notin B$, since if $A\subseteq B$, then $B$ may very well contain elements that $A$ does not contain.  

Similarly, suppose we have that  $p\rightarrow q$. If $p$ is true, then it must be the case that $q$ is true.  But if $p$ is false, that does not necessarily mean then that $q$ is necessarily false. (For all we know, perhaps $q$ is true regardless of whether or not $p$ is true.) So $q$ can be true, while $p$ is false. 

I don't know if this analogy helps or not. But it was the above analogy (correspondence) that helped me to firmly grasp the logic of material implication.



Here's a more down-to-earth example you may have already stumbled upon: 

CLAIM: "If (it rains), then (I'll take an umbrella)":

I'd be lying (my assertion would be false) if (it rains = true), and I do not (take an umbrella).

But perhaps it's cloudy out, and I decide I'll take an umbrella , just in case it rains. In this case:

If it doesn't rain (it rains = false), but I took my umbrella (true), my claim above would not be a lie (it would not be false).
Sure you can have a valid "implication relationship" between two errant pieces of nonsense.

If we have M=>H between moon and hamster populations then the truth table is 
M=>H == not(M and not H).  In modal logic (many-world) terms this means we allow all worlds to be possible EXCEPT the one that has a cheesy moon and at least one non-green hamster. 

This is the main effect of the implication relationship, it acts as a prohibition on one case, indeed this is the essence of an asymmetric dependency. M=>H means M depends on H in the sense that we cannot have M without H. If H is false then M cannot be true. Asymmetric dependency relationships are very common, hence the utility of this connector. 

In the other direction it matters not, there are at least two worlds that have a normal moon, and in only one of them the hamsters are all green. You are NOT forced to live in that world!

The Wason selection task is a good illustration of utility of material conditional support for deductive reasoning:

https://en.wikipedia.org/wiki/Wason_selection_task