Section 3 of Glanzberg's paper gives an overview of the ways in explicit and common-or-garden-contextual restrictions on quantifiers work (as background to a discussion in later sections about how "background domains" are fixed). This section isn't intended to be more than just a quick set of reminders about familiar stuff, so we can be equally speedy here.
Going along with Glanzberg for the moment, suppose the "background domain" in a context is M (the absolutist says there is ultimately one fixed background domain containing everything: the contextualist says that M can vary from context to context). More or less explicit restrictions on quantifiers plus common-or-garden-contextual restrictions carve out from this background a subdomain D (so that Every A is B is interpreted as true just when the As in D are B, and so on). How?
Explicit restrictions are relatively unproblematic. But how is the contextual carving done? There are cases and cases. For example, there is carving by 'anaphora on predicates from the context', as in
1. Susan found most books which Bill needs, but few were important,
where 'few' is naturally heard as restricted to the books that Susan found and Bill needs. Then second, there is 'accommodation', where we rely perhaps on some Gricean mechanisms to read quantifiers so that claims made are sensible contributions to the conversational context. For example, as we are about to leave for the airport, I reassuringly say that, yes, I'm sure,
2. Everything is packed
when maybe some salient things (the passports, say) are in plain view in my hand and my keys are jingling in my pocket. Here, my claim is heard, and is intended to be heard, as generalizing over those things that it was appropriate to pack, or some such.
There's a third, rather different way, in which context can constrain domain selection, that isn't a matter of domain restriction but rather a matter of how, when an object which is already featuring prominently enough as a focal topic of discourse at a particular point, "we will expect contextually set quantifier domains to include it". (Though I guess that this point has to be handled with a bit of care. The taxi for the airport arrives very early: we comment on it. "But," I say, "we might as well leave in the taxi now. Everything is packed" The quantifier of course doesn't now include the taxi, even though it is the current topic of discourse.)
Well, so far, let's suppose, so good. But how are these reminders about common-or-garden contextual settings of domains going to help us with understanding what is going on in fixing 'background' domains? The story continues ...