Actually I wrote that article on cp-algorithms. (Well, I translated it from e-maxx.ru). I'll try to add a few nice pictures and an extended explanation in the article tomorrow.

For now only something very short: During the construction of the trie you have marked all vertices that correspond to the end of a pattern and stored which patterns end in each vertex. Currently you are at vertex v (e.g. after processing the first x characters of your text). And you want to know all patterns end exactly at this position. You can find them by doing the following:

while (v != root) {
    print pattern ending in v
    v = suffix_link(v)
}

It should be obvious that this prints all patterns. A pattern that appears at that location is a suffix of the text and by following suffix links we check exactly all possible occurences.

Take a look at the following Aho-Corasick trie for the pattern aaa, aab, and a (red are positions where pattern end, blue are suffix links):

You currently processed the text xyzaaa and are at vertex 3. You print the pattern aaa, follow the link to 2, follow the link to 1 and print a and follow the link to 0. Notice that we visited 2 but actually accomplished nothing at that location, since no pattern ends at that vertex. Exit links are just suffix links that skip such redundant vertices. The exit link from 3 would point directly to 1. And therefore you are guaranteed to find all ans pattern that end at that position in the text in O(ans) time.

My implementation of Aho-Corasick is here: Github Notice, I don't compute the array go on the fly, but instead do a BFS after adding all strings, during which I generate all suffix and exit links (here called next_terminal).

For reference, I learned the BFS approach from here.