The colorful fleshy part at the base of the puffin’s bill is called a rosette. I wrote a bit about it in my post “I *SO* LOVE PUFFINS“. These fleshy little rosettes are not only used for attracting mates but they are also a way the puffin can open its beak wider and in both ways, rather than just flat open like other birds. Here, let me show you. In the picture below, do you see how this puffin can open its bill much wider, as if it gives the jaw a wider range to allow the puffin to open its beak more—to hold and carry more fish—to communicate better. I am certain there are many benefits of having the flexibility to open its bill wider.
As opposed to other bird beaks which open only so far at the base and mostly get wider towards the tip of the beaks as the birds open their beaks; puffins can open their bill at the base as well, thanks to the fleshy rosettes, and then open wider going outwards toward the tip, giving the puffin a much larger range of openness, more space and flexibility within its bill.