Personally, I don't like fishing and I believe that if you aren't going to eat the fish it's cruel, but I was playing Stardew Valley and I had been doing a lot of fishing when I started thinking about the physics of fishing.
If a fish is on the hook, you start to reel in the line. As you do this, you created tension in the line. If the fish is able to produce a larger force in the opposite direction of this tension, the line will unroll a bit more. However, if the tension in the line is creating a bigger force than the fish can exert in the opposite direction, you will be able to reel in the line more and the fish will accelerate forward.
While doing this, you are also working against the force of friction from the fish and water, so you have to be careful not to snap the line by creating too much tension.