Think of the old expression eyes are windows to the soul. This is especially true with animated characters because there is only so much detail any animator can reasonably draw, though some have tried their hardest. In anime, the mouth is used to enhance the expression in the eyes. Another way to think of it is like writing a sentence. The eyes are the words and the mouth is the punctuation.
Big eyes can also have something to do with the relative age of a character.
Although this is not always true, younger characters are often drawn with larger eyes. Since many anime scripts contain protagonists that are in their teens and early twenties, those characters would naturally have slightly larger eyes than someone in their thirties of forties. The consistency in this is largely dependent upon the style of an individual production studio and is not a steadfast rule. But either way, statistically it is pretty common. Many anime characters have elaborate and excessive hair.