Grothendieck topology for a non-small category
We need the category $C$ to be small so as to define the functor category $Cat(C^{op},Set)$, i.e. the category of presheaves, so that it is locally small.
The trick works with the category of manifolds because the category of smooth Hausdorff second countable manifolds is equivalent to a small category (use the embedding theorem, so that every such manifold is embedded in a high-dimensional $\mathbb{R}^n$ - there are only a set of such submanifolds if we take them to be literal subsets of Euclidean spaces). People generally don't want to deal with non-Hausdorff manifolds (some do) and almost never non-second countable ones (only show up as pathological counterexamples, for example the long ray). So one can repeat this trick for any large category which is equivalent to a small category.
There might be a proper class of bases for any given topology on a large site. It is entirely equivalent to work with a pretopology/basis rather than a topology, as one doesn't need the full topology to define sheaves (or possibly stacks etc), which is the whole point. Once you have the topos of sheaves, the choice of basis is immaterial, though sometimes convenient for calculations.
The real thing you want is that the axiom WISC is satisfied, especially when the definition of a pretopology (=basis) doesn't demand that each object only has a set's worth of covering families. Consider for example the category of groups with the pretopology consisting of single epimorphisms as covering families. This is a large category such that each object has a proper class of covering families. But one can find a set of epimorphisms such that any epimorphism is refined by one in that set, and this is the statement of WISC. For a small category, WISC is automatically satisfied, and also when the definition of basis is such that only a set of covering families is associated to any object.
A non-example is the category of schemes with the fpqc topology, and so there we cannot sheafify an arbitrary presheaf. Another non-example can be the category of sets if one assumes the negation of the axiom of choice (independent work of van den Berg, Karagila and myself).
Re 2:
In my opinion, none of the answers just yet have hit the nail on the head about "why this trick works". The real reason is Urs' comment about dense subsites. Whether one takes manifolds to mean 2nd countable + Hausdorff, or whether one removes these conditions and considers all topological spaces with a smooth atlas, the topos of sheaves over their corresponding sites are equivalent. This is why the trick works; it's not simply that you don't care about pathological manifolds (it's actually very convenient to have them around when you're talking about differentiable stacks), but that you don't need to include them in your site of definition, because they are faithfully represented by the sheaves they induce over any full subcategory of manifolds, containing at least one manifold of each dimension. E.g., one can consider the full subcategory of manifolds spanned by only those of the form $\mathbb{R}^n,$ and takes sheaves on this site, and this is equivalent to taking sheaves on the site of all manifolds. The latter site is NOT essentially small, but it doesn't matter. In full generality, you need a site with a "small set of topological generators": see SGA4 (in particular expose ii, theorem 3.4) and the link Urs posted http://ncatlab.org/nlab/show/dense+sub-site
The other answers are all good, but I thought I would also point out that one doesn't have to require that sites be small, or have small dense sub-sites, or satisfy WISC. I think one does generally want to assume that each covering sieve contains one that is generated by a small family, and also an additional "solution-set condition" that is vacuous for small sites, which I studied in this paper.
In particular, I showed there that there is a reasonable notion of "the category of (small) sheaves" on such a large site, and in most cases it does admit a "sheafification" functor from the category of (small) presheaves. What's different is that sheafification doesn't necessarily have a right adjoint, so that "sheaves" can't necessarily be identified with particular presheaves. The "category of small sheaves" is generally not a Grothendieck topos, nor an elementary one, but it satisfies all of Giraud's axioms except for the existence of a small generating set, and it has "the same" universal property (in a certain sense) as the topos of sheaves on a small site.