First, equip a fishing rod and bait in your ranged and ammo slots. Then, find a body of water, place yourself at the edge, and type /fish, or choose Fish from the menu.
Wait a few seconds for a bite. When a bite comes, the fish will appear in your target window. The rod will pull to the left and the right. If your camera is facing the back of your character, move the directional buttons in the opposite direction the rod is pulling. You will notice this lowers the fish's stamina (its HP bar). Once the fish's stamina reaches 0, hit enter to reel him in.
Other interesting facts about fishing:
Hitting Esc at any time will cancel your fishing attempt. You will loose any bait you are using, but not lures, and you will not break your rod.
When your fishing instinct has three "!!!" instead of "!" when you get a bite, that indicates you have a monster on the line or a large fish. If you do not wish to fight the monster it would be a good time to hit Esc.
You can still pull in a fish before depleting its stamina to zero, but you usually need a high skill to do this.
When the rod is straight, move the directional buttons downards to keep the to keep the fish from regaining strength.