$E_L(n)$ describes how many bits of information you get from a string of length n, assuming it belongs to the language.
For many languages the function would be a rather smooth function in n - there are of course languages where the language only contains strings of even length, for example, in which case $E_L(n)$ isn't even defined for odd n.
$E_L(n)$ together with the logarithm of |∑| tells you how likely it is that a random string is in L. Usually people try to create strings in a language (that's what programmers usually do), and a low entropy means small mistakes are more likely to get caught.
Which is a good thing, because usually a string in the language L has some semantics, and when I wanted to create a string that is supposed to be in L and has semantics X, and a make a small mistake using one or two wrong symbols, I want to be told that my string is not in L, and not produce end up with a string in L with the wrong semantics.
If a language has very high entropy, so that most random strings are in L (but with probably very different derivations), it is very hard to find errors because anything could be in L.