Q:

What is the utility, exactly, of the Deduction Theorem? 

The Deduction Theorem states that for a set of assumptions $\Delta$ and two wffs $A$ and $B$, we have the metalogical relationship:
$$\Delta \cup \{A\} \vdash B \implies \Delta \vdash A \to B$$
In other words if we can prove $B$ from some set of assumptions (conjoined with $A$), then it's the same as proving $A \to B$ from our assumptions. 
My question here is not asking for a proof: I'm asking what it's allowing us to do.  I'm not even fully sure I understand what this is saying or how its making our lives any easier. 
From my uninitiated perspective it would seem that the advantage is letting us treat the logical connective $\to$ as a way to represent a proof from $A$ to $B$ within our logic system rather than mucking about in the metalogical realm, but I feel like I'm looking at it wrong / interpreting this result incorrectly. 
What is the deduction theorem telling us?  How does it make things more "natural"?  What is it doing? 

A:

The theorem says that to prove an implication it is enough to assume the hypothesis and proceed to prove the conclusion.  Proofs of that kind tend to be more natural than proofs that conclude the implication directly.  
Just as in regular mathematical practice: many theorems have the form "Assuming $A$, then we have $B$", and we usually prove them by assuming $A$ and using it along the way to conclude $B$. And we feel we are done, even though the task was not to prove $B$, but instead "$A$ implies $B$". 
So the theorem is saying that the rules of propositional calculus "capture" a common way of reasoning in practice.  The more of such rules we manage to establish the easier it becomes to use the calculus to prove statements.  The first few formal proofs always feel strange, perhaps even artificial.  These results, such as the deduction theorem, aim at turning this rigid, "artificial" proof system into a natural and useful tool. 
There is another key reason why the theorem is useful, now in the context of propositional logic itself, namely, it is a basic tool that helps us prove the completeness theorem.  We establish other metatheorems along the way (such as: We can argue by contradiction), and all combined help us in making feasible the task of showing that the proof system is complete: We define a notion of truth for propositional formulas (via truth tables), and completeness says that the proof system is enough to deduce all true implications.  This is a very useful first step in more elaborate contexts where we may want to mechanize proofs in more complicated logics. 