How do model categories assist in localizing at the weak equivalences? I am interested in localisations of categories with weak equivalences and in particular in localisations of model categories at their weak equivalences.
Let $\mathcal{C}$ be a category with weak equivalences. In this note (page four), David White explains how in trying to define the morphisms in the category constructed by formally inverting weak equivalences (i.e. localising at the weak equivalences), we obtain that the morphisms between two objects are "zig-zags" of morphisms in $\mathcal{C}$. White then writes that for any two objects $X Y$ of $\mathcal{C}$, these morphisms do not necessarily form a set - even when $\mathcal{C}$ is the category $\mathbf{Set}$, they form a proper class.
He then says the following

Attempting to get around these set-theoretic issues leads you to model categories.

I am interested in how one is lead to the idea of a model category, as White explains, and also (broadly the same question) how the structure of a model category actually solves this problem.
 A: The problem is that the localisation $\mathcal{C}\rightarrow\mathcal{C}[\mathcal{W}^{-1}]$ is not in general calculable by either left- or right- fraction, but is a mixture of both. This last sentence is more or less quoted from the introduction of Quillens Homotopical Algebra. 
I cannot make all the relevant defintions in this post, so if you have not seen these terms before I would suggest Borceaux's Handbook of Categorical Algebra Vol I: Basic Category Theory, $\S$ 5, for the details. You might also try Gabriel and Zismann's Calculus of Fractions and Homotopy Theory, which was Quillen's reference book for the original theory. 
The point is that given the weak equivalences $\mathcal{W}$ you are lead naturally to look for certain reflective subcategories $\mathcal{C}'\subseteq\mathcal{C}$ for which the localisation $\mathcal{C}'\rightarrow \mathcal{C}'[(\mathcal{W}\cap\mathcal{C}')^{-1}]$ is calculable by either left- or right-fractions. These are your subcategories of cofibrant and fibrant objects, respectively. The fact that these localisations are calculable by left- or right-fractions means exactly that the resulting localisation is a locally small category. 
Then the idea of the complete model structure at this point is now to guarantee that the induced maps $ \mathcal{C}'[(\mathcal{W}\cap\mathcal{C}')^{-1}]\rightarrow\mathcal{C}[\mathcal{W}^{-1}]$ result in an equivalence of categories. This is exactly Quillen's Theorem 1 in his book. Once you understand from Borceaux what is going on you see that it has been the factorisation systems required by the model structure that have played a prominent role in the outcome. Clearly the full structure of a model category is really little more than a framework to make these gadgets work.
Finally I'll give my own opinion, that I'm not sure the set-theoretic issues lead you directly to the full structure of a model category. Rather, what Quillen's theory provided was an elegant solution to the problem, that although might seem too highly structured to be relevant, is actually available in most cases of interest. 
I would suggest reading Dwyer , Kan, Hirschhorn and Smith's monograph Homotopy Limit Functors on Model Categories and Homotopical Categories for their (expert) thoughts on the problem from a more general perspective than model categories. 
