How you store that freshly caught fish will depend on where you are fishing. If you are on a boat, there will usually be provision to store fish within the boat. The main rule to remember is that it must be kept as cool as possible to prevent it from causing food poisoning when eaten.
If you are fishing in a river and plan to stay in one spot, it is possible to keep the fish alive in a burlap bag by making a pool of rocks or sand to place the bag in and keep it anchored with a heavy rock. This will keep the fish fresh for later in the day when you have time to kill and gut it. Make sure the pool you make is separated from the main river by a firm wall of rocks or sand, so there is no chance of the fish escaping should it wriggle out of the bag.
You can also place it in the bag and tie the top tightly with twine, then sink the bag into shallow water. Tie it to a low branch or a tree near the edge of the water with another strong cord. This is a good idea if you plan on moving up and down the river. Your catch won't be immediately visible to others who might want to help themselves.
Some people keep their live fish in a bucket of water on the riverbank, but this has its dangers because fish can jump out and wiggle back to safety very quickly - and if children are playing nearby there is the risk they could knock the bucket over, thus allowing your catch to escape.
If you prefer to kill the fish straight away, you'll need to have a cooler with ice in it to keep it, otherwise it could be bad by the end of the day. Gut the fish and place it on the ice, but be sure it doesn't float in the melting ice water. Once you have the fish home, cleaned and scaled, wrap it in a plastic bag and keep it in the fridge until you are ready to eat it. If you don't plan to eat it by the next day, the freezer is the best place to keep it. Fish will keep for some months in the freezer.