I'm talking about books on specific academic topic, e.g. history of the Aztecs before colonialism, which are published by university presses like the Oxford University Press. How expert are the editors of such books? Can we count on them to ensure that authors don't misrepresent their sources, or don't make statements that are known by experts in that specific area to be clearly false? Or is their expertise really minimal and we must only trust on the authors?
The procedure (and quality) of contributions to an academic anthology will differ with the type of collection.
Some books collect the written versions of presentations held at symposia. Everyone who was invited to speak at the symposium will be published in the book, and there is no screening for the printed publication. So if you manage to fake yourself into a congress, whatever you presented there will make it into print. This is understood by readers of an anthology documenting a symposium.
Other anthologies collect articles that were written specifically for that publication and on invitation by the editor. For example, a group of scholars may want to document the state of their field and will invite other scholars in their field to write about their research. These other scholars will be people the editors know, often personally, and they are of course intimately familiar with the reasearch of those scholars (or they wouldn't invite them to contribute). Scholars aren't usually chosen because they all agree on the matter, and often there is some intense dissent in these anthologies (which is a good sign for the neutrality of the editors). In an invitation-only anthology you can be sure that the editors carefully read all the submissions and, if necessary, ask for revisions or even reject a submission, if it doesn't meet their requirements. Certainly not all editors have the same level of expertise, and the overall quality of the publication will depend on the quality of the editors, just as in a journal, not all of which have the same level of quality, either. If you are an insider in a field, you will know whom you can trust and whose publications – anthologies and journals – you have to be wary of.
Finally, some anthologies are handbooks or teaching textbooks of a field. These will have the most careful editing process, as they are meant to reflect on the reputation of the editors. Usually in these books the editors have a concept regarding structure and content and the contributing authors have to subordinate to this. Often the authors are recruited from the editor's research partners or research team. The quality of these books depends totally on the editors, but is usually very high. Textbooks can be bad, of course, but you can usually trust their content.
As for how editors are chosen, there are two ways a scholar can become an editor: either he or she comes up with an idea and approaches a publisher; or a publisher has an idea and approaches a scholar. In both cases, the publisher will attempt to recruit an editor who has a high level of reputation and expertise in their field. Often the publishers have series editors who are themselves experts and familiar with the researchers of their field. Since success as an academic publisher depends on the respect and acceptance of their audience, that is, the scholars of a given field, publishers will do everything in their power to select editors that turn out quality work. Publishing books that turn out to misrepresent a science will mean the death of a publisher.
All in all you can generally be sure that editors of academic anthologies (and journals) know their field and try their best to present good information to their readers. Nevertheless mistakes happen, and there are black sheep, so you should read with a critical mind and, if possible, check critical information agains other publications or, if you can access it, the original data. This is, of course, how you should always read, as a scientist.