This is a real phenomenon, which I've had to deal with in practice as an editor. It's by no means universal, and it's probably a cop-out to some extent, but in at least some cases it genuinely indicates that standards are temporarily higher than normal while the journal deals with a backlog.
My impression is that it is caused by several factors:
A publisher that targets a certain number of pages per year and will not substantially exceed this target (on the grounds that it will cost more without raising subscription revenue correspondingly), regardless of how many papers the editors accept. If there are too many accepted papers, then they will be published gradually over time.
A decentralized editorial board in which individual editors recommend papers for acceptance without having to justify their decisions in terms of the publication rate of the journal.
A sufficiently large backlog that authors start to get upset about how many years they have to wait to see their paper in print.
All of these factors can contribute, but they are not all necessary. For example, Journal of the AMS has had backlog issues in the past, despite having an editorial board that makes decisions jointly.
Typically what happens is that the publisher or the editor in chief starts to get worried about the high backlog (since it can hurt the journal's reputation) and pressures the other editors to accept fewer papers until the backlog starts to decrease. During this period, it can be tricky to coordinate with referees on the new standards, so editors typically desk reject papers they aren't excited about, even if those papers might plausibly have been accepted normally.
Each year the AMS publishes backlog estimates for mathematics journals (here is the 2015 version). If you see a particularly high backlog listed there, then the stated reasons are probably real. If you don't, then the editor might be trying to soften the blow of rejection, or it's possible that the AMS's backlog estimate is not right.
It's worth noting that a small backlog is good, since it evens out the random fluctuations in the acceptance rate and keeps the publisher happy. It's only a problem to the extent that it starts to upset the authors, and it has been less of a problem in recent decades (since the AMS started publicizing the backlogs).
I haven't in my admittedly limited experience heard similar wording in rejections from, say, experimental sciences journals.
Yes, this phenomenon does not seem to exist in most other fields. In mathematics, formal publication is not the primary mode of communication (rather, it's for permanent archiving), so nobody is too horrified by backlogs, and this lets things get out of hand occasionally. In the experimental sciences, authors demand rapid publication, so there is no concept of accepting more papers than you have space to publish and developing a serious backlog.