You can cast a line by pressing "CAST" button on the lower right corner.
After doing so a fishing minigame will start, in it, you have to manipulate the reel to keep the energy bar from fully filling and emptying. By keeping the power level in the smaller window you will earn combo points. Your goal is to diminish the fish stamina points (represented by the green bar with hearth icon) every fish has its own stamina - bigger and rarer fish will have more stamina.
The more combo points you'll get the more points you will earn for catching that fish.
Fully filling the combo bar (up to 100%) will result in 5% more points.
Pressing it on weight icon will bring up "how to catch a bigger fish" window:
Coins icon will show your current information on your bonus coins. Every day you can earn coins for the 50 first caught fish. After that, you will earn 1 coin for every fish.
After successfully catching a fish you will see all information about your catch:
Size, rarity, weight, earned points and coins. By pressing "Claim" button points and coins will be added to your account.