Fishing is a QTE mini game found at jetties sprinkled throughout Middle-Earth. Fishing spots can be found both in levels and on the world map; they generally contain a key item for advancing the plot or unlocking a Mithril brick. A fishing spot can be found directly within Hobbiton on the west side of the river. Each spot can be continuously fished, though you need to reload the area before exploiting this.
The game gives you two fishers as part of the story-related unlocked characters: Merry, who fishes with a rod, and Gollum, who fishes with his hands. Merry is available after The Black Rider (which also has a fishing spot). Gollum doesn't unlock until much later in the game. In addition, there are two Treasure Trove items which allow any character to fish: the Fishing Rod and the Mithril Fishing Rod.
To fish, select a character with a rod, approach a jetty, and press B. you'll then need to to make three button presses to complete the mini game. These are random, but are always either X or A. Each press is timed, with a shrinking circle about the button icon. The achievement description is a bit vague, as the button icon pulses before the circle touches it. What you want is to press the corresponding button once the circle connects. At this point, light beams will strobe out of the icon. If you get all three presses correctly timed, you'll be rewarded with studs.