Peer review is mostly a requirement but some have professional editors as well, which is a paid position. Costs for OA journals are almost always borne by those successful at getting through the process which means that any costs are related also to the rate of rejection. For journals groups like Nature, that may mean an OA cost paying ~10-20x the actual cost (plus profit margin) due to a high rejection rate.
Broadly the reason this has stuck around so long is that what publishers are assigning to journals is a vague notion of trust. They are incentivised to keep "high quality" articles in their list as that's what people want to be listed in. The recent discussions around reproducibility tie in with discussions around clickbait news, in that the incentive of citations is not identical to the incentive of quality.
Peer review isn't free to do either despite one part of the labour being free, PeerJ increasing their costs is a good overview of this.
(disclaimer - work for a company (digital science) that's related to publishing in that we analyse and combine the data, and we're owned by a larger publishing house (holtzbrinck) but I have no horse in this game myself)