You must be reading a very old book. The word "anon" is archaic and never used anymore. It means "soon" and is from Old English. These days, you might find it in poetry or in Shakespeare, or some other old literature. You also might hear it if someone is trying to imitate old speech (usually as a joke). You don't need to know how to use it because you will sound VERY strange if you do. You only need to know what it means so that you can enjoy old literature.

Be careful using this word, because these days it's mainly an abbreviation for 'anonymous'. The old meaning is only known to very literate people.

It means "soon", "later", or "from time to time". Rather than call it "archaic", I would classify it as an "archaism", a word that is survives in spite of obsolescence because it is useful in everyday speech for its style, emotion, or cultural weight. The fact that it will remind educated people of Shakespeare and great poetry lends it eloquence. Other useful archaisms are "thou, thee, thy, thine", "behold", "hark", "lo", "naught", "ere", ... (and many many others. Ask AI for a list of "archaisms".). These are classy words. If you learn how to use them well you will sound truly awesome.