Fish are used to craft teleportation tapestries and some pet snacks. At level 125, you'll need to do some fishing in order to craft the Revered Want, which gives extra energy (and fishing luck). This wand is most useful to those who garden, fish, or train pets (and/or any combination thereof) and who therefore would like more energy.
Fishing chests may contain extremely rare and unusual gear, housing items and/or pets. Retired code wands are available from fishing chests you find in Zafaria's Elephant Graveyard, for example. In Avalon, you can find rare paintings, trees or other unusual housing decorations not available anywhere else in the game.
Catching chests and selling the fish you catch can bring in quite a lot of gold, though at the expense of energy you otherwise might have used for gardening or pet training.
Fishing is also completely optional. I have one or two wizards who do the fishing for crafting and decorating purposes, and the rest don't bother.
If you are a badge collector/achievement hunter, you will need to do fishing to obtain those badges. Also, certain crafting recipes require fish for their ingredients.
I like fishing for the crafting but most of all I liked the sense of achievement of catching them all, using them to craft aquariums then displaying them. I've also had some really lovely drops from the fishing chests - pets, mounts, tapestries, housing items, wands etc.