How to make sense out of the $\epsilon-\delta$ definition of a limit? The informal intuition for the limit of a function is this:

What is the value of the function $f$ as $x$ gets infinitely close to $c$?

How on earth does this monster
$$ \lim_{x \to c} f(x) = L  \iff  (\forall \varepsilon > 0)(\exists \ \delta > 0) (\forall x \in D)(0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon)$$
capture our intuition?
I want the answer to explain and motivate every bit of this formal statement. Remember: I am able to parse the statement, I just don't see how it captures our informal intuition.
 A: The definition capture not an open question as 

''What is the value of the function $f$ as $x$ gets infinitely close
  to $c$''

but the exact statement:

$f$ can be as close as we want to $L$ if $x$ is sufficiently close to
  $c$.

If you think to this you can see that the ''monster'' works well.
A: The definition can be read, in human words:

For every positive $\epsilon$, there exists such a positive $\delta$ that if $|x-c|$ is smaller than $\delta$, but larger than $0$, then $|f(x) - L|$ is smaller than $\epsilon$.

First of all, lets's get the double inequality out of the way. Basically, $0<|x-c|$ is just saying that $|x-c|$ is not equal to $0$ (since it can't be negative), and saying that is just saying that $x-c$ cannot be $0$, or in other words, that $x$ is not allowed to equal $c$.
Now to a non-mathematician, that still makes very little sense, but take into account that $|a-b|$ is really the distance between numbers $a$ and $b$.
So, we can translate the definition into

For every positive $\epsilon$, there exists such a positive $\delta$ that if $x$ and $c$ are two distinct numbers and the distance between them is smaller than $\delta$, then the distance between $f(x)$ and $L$ is smaller than $\epsilon$.

But that still does not ring quite "natural" But what does the "for all $\epsilon$, there exists a $\delta$" in the beginning mean anyway? Well it means that whatever $\epsilon$ you give me, I can find a $\delta$ such that the condition will be true. So:

No matter what $\epsilon$ you choose, I can find such a positive $\delta$ that whenever you take any $x$ near (but not equal to) $c$ that is less than $\delta$ away from $c$ which are closer together than $\delta$, $f(x)$ will be closer than $\epsilon$ away from $L$.

Getting warmer to something readable? Well, let's get rid of the variables even further:

No matter how close you want $f(x)$ to be to $L$, I can tell you how close to $c$ you need to pick your $x$, and if you pick your $x$ that closely, then I can guarantee that $f(x)$ will be as close to $L$ as you originally wanted it to be.

This is very similar to what MPW wrote in comments:

$f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$

A: It means that we can make $f(x)$ take on a value as close as we like to $L$ (closer than any positive $\epsilon$ that we choose) by bringing $x$ near enough to $c$ (nearer than the positive $\delta$ which is asserted to exist in the definition).
A: Imagine that you want to be $\epsilon$ close to to limit $L$, than the definition just says that you can pick arbitrary any $x$ from some $\delta$ neighborhood of $c$ and with $f(x)$ you are $\epsilon$ close. 
A: Intuitively it makes sense if you read the right-hand side as saying "it is possible to get $f(x)$ as close to $L$ as you like by demanding only that $x$ get very near $c$".  If the right-hand expression is always true, then the limit on the left will be true.  This does not imply that $f(c)=L$, so $f$ is not necessarily continuous [that would be something like  $\lim_{x->c}f(x)=f(c)$]. Finally, to gain an intuitive feeling for your monster take a function for which the left-hand limit fails [eg., $f(x)=0$ if $x<0$, $f(x)=1$ otherwise], then [with $c=0, L=1$] try to get the right-hand side to work [ie., given $\epsilon=0.1$ try to find a $\delta$ which works].  Do this and your monster will make sense.
