Denis-Charles Cisinski has a beautiful book called Les Préfaisceaux commes modèles des Types d'Homotopie, which gives a very very powerful framework for building model structures on presheaf categories (and more generally Grothendieck toposes), and after building up this framework, the model structure for simplicial sets drops out literally for free.
Here's a link to it from his website: link.
He also proves some nontrivial conjectures of Grothendieck that are important for derivator theory, among other things. Rick Jardine published a summary paper of this book, which is also worth reading.
Note: The framework is built up entirely in chapter 1, so even if you don't want to read the whole book, the first chapter is what you need.