Fishing is accomplished by using the fishing rod from your pack. Simply cast your line and reel it back in. The longer you have your line in the water the more likely you'll catch fish. You can also unreel when you hear a bubble or splash sound, which is a very likely time you'll catch a fish while also being more likely to catch a larger fish.
As your fishing rod is in the water a value called chance goes up. When you unreel a random number is generated. If your random number is less than chance value, you'll catch a fish. In other words the longer you are fishing the more likely you'll catch a fish because of a higher chance number.
Is is technically possible to catch a fish really quickly by throwing your line and reeling it back in often, but the chance is low... so it's probably worth it to fish for a bit longer if you aren't catching anything.
It's generally good though to unreel when you hear a water splash sound, as it's very likely you'll catch a fish then.The fish caught during the sound cue is probably more likely a larger fish as well.