In words, lim (x->c) f(x) = L means for any number x in your interval, the closer x is to c, the closer f(x) will be to L.
Note |x-c| just denotes the distance between x and c, or the absolute value of the difference.
So saying "for any e>0 there exists d>0, any for any x, 0<|x-c|<d -> |f(x)-L|<e" means
For ANY small distance e, if the distance between f(x) and L is less than e, than you can ALWAYS find a d so that the distance between x and c is less than d.
Hope this helps! It easier to explain with pictures.