Peer review is done by fellow researchers, without being paid by the publisher. However, the publisher does find the editorial board (and sometimes pays them) that finds the peer reviewers, often (but less and less so) provides typesetting and proof reading services, and most importantly provide the infrastructure that makes the articles available and makes sure that they remain available in the future.
(And there's a ton of other things, like making sure people pay to access articles when they have a subscription and the like, but those are services they provide to themselves...)
For well-established journals, the editorial board finds new editors, so there is no cost involved for publishers.
With respect to typesetting and proofreading, my personal experience is that the 'service' provided by Elsevier was of such a low quality that it actually caused _increased_ effort on the author's side.
Making the articles available is indeed the core business of a publisher. However, prices for that kind of service should have decreased significantly over the last two decades, not increased.
Yes, to be clear: I am not arguing that the value they provide justifies in any way the extortionate fees they're asking. Quite the opposite, in fact - I believe it's mostly "justified" (or at least made possible) through a disfunctioning market.
(And there's a ton of other things, like making sure people pay to access articles when they have a subscription and the like, but those are services they provide to themselves...)