Why is peer review anonymous?

The big reason for anonymous peer review is to allow the reviewers to speak their mind without fear of the authors taking revenge. Even if it's easy to identify, retaliation can happen: it's already decently common for authors to get angry when their paper is rejected for reasons they do not agree with (example). Even if the authors don't retaliate this obviously, future relations with the reviewer can easily be strained: for example, say the author and reviewer communicate several times and eventually the author decides that the reviewer's an idiot and submits elsewhere. If the author later finds themselves reviewing the reviewer's grant proposal, it will be hard for the author to remain objective.

Another issue with non-anonymous peer review is, the signature of a senior person as a reviewer can carry the implied threat of "do what I say or I will crush you". The consequences can be unpleasant for the authors, especially if they're junior. Example.

Yet another issue with non-anonymous peer review is that it's not clearly advantageous. There is no strong evidence that open peer review improves review quality (or makes it worse for that matter). Instead there was an immediate negative effect in that more people decline to review. If it's the case that there's no advantage, then implementing a fix seems rather unnecessary.

If I review some professor's science paper and it's poorly controlled garbage, I either accept it anyway or I have a new enemy. Now I have to deal with the fact that submitting papers to all the journals that they're editors for will go poorly for me, and submitting grants that they'd be part of study sections for will be a waste of time for me.

Even people who don't have to worry about their own careers need to worry about this, because big name scientists fight by trashing each others' students' careers. Do you really want to have some promising junior scientist who worked for you get repeatedly shit on because you reviewed a paper?

Without scratching below the surface it seems anonymous peer review is the only way to maintain standards of quality and honesty. Not everyone who is a qualified reviewer is going to be 100% comfortable being openly critical of their peers' work; if the integrity of the results suffers even slightly over time due to that it is reason enough. On the flip side, the reviewers without any qualms about brutal honesty will see less difference btwn sugar coating and tactful delivery, which may result in an unfavorable imbalance towards more harshly toned reviews on average, and thus potentially more discouragement than encouragement to the reviewees.