You should have enough Cat Shines after completing Scamper Shores to activate the Lakeside Giga Bell for the first time. It requires five Cat Shines and whenever Fury Bowser wakes up, so will the Giga Bell. To start the boss fight with Fury Bowser, simply head over to the small island with the Giga Bell and collect it like any other normal power-up. This will cause Mario to basically turn into a giant cat, about the same size (or so) as Bowser.
The first fight with Fury Bowser is pretty simple, but you will need to remember that attacking him like a normal enemy isn’t going to work. With that in mind, it’s best to keep your distance with Bowser at the start of the fight. The first thing he will do is jump to a random spot, then queue up a fire breath attack, similar to the one needed to break the Fury Blocks. Wait for him to sweep the fire breath your way and if it’s going to hit you, simply jump over it. After the fire breath, Bowser will retreat into his shell, then jump into the air.
You will see a red marker on the ground at this time, which is where Bowser will land, of course. The easiest solution is to keep moving and Bowser should always miss you. Upon landing, though, a shockwave will be send out, so you will need to jump over it to avoid taking damage. Sticking close to Bowser here is the idea, since he will be upside down when he lands, making him vulnerable to a Ground Pound. Do this on Bowser and you will take out a good chunk of his health bar. Depending on how often you collected Shines when Bowser was out, it will take multiple hits to deplete his health all the way.
Note that should you get hit, then Mario will still stay giant, but will revert to big Mario. Like normal, getting hit once more will make him small Mario (meaning no power-ups at all). The Giga Bell does respawn, if you prolong the fight enough, but the timer is pretty long and almost requires you to not go after Bowser. Once Bowser’s health has been depleted all the way, he will go into hiding again and more of the map will open up.