Epsilons and deltas can be pretty overwhelming when you first experience them. Epsilon (e) and delta (d) are just error bounds. You can think of them as small positive real numbers.

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.