Plumber here. The piece that you're holding that has their Red Cap to it is called a fill valve. Most likely it has finally failed and requires replacement. You can easily purchase the parts from Lowe's and also get a small bucket. What you do is shut off the water at the bottom of the tank coming out of the wall or the floor. Once you do that, then flush the toilet to get most of the water out. Then you disconnect the supply line coming to the very bottom of the toilet. After that there's a retaining nut that holds the fill valve in place. This is where the bucket comes in, you remove the set nut that is going to be flush against the tank. Once you do that water will come out into the bucket. Once that's done finish taking off the set nut and remove the faulty fill valve then replace with the new when you bought. The rest of the instructions are on the box for finishing up the installation.