There are multitudes of blog posts and articles about signing reviews. Two sources I find particularly useful include multiple viewpoints from different scientists (from The Molecular Ecologist) and interesting data about frequency of signing in different subdisciplines (from publons).
Many of the arguments in favor of signing reviews refer to improving the quality of the scientific assessment in the review, as well as keeping the tone of the review civil and constructive (as opposed to possibly derogatory). I don't believe the content of my reviews differ based on whether I sign my name to them, because I try hard to be rigorous (and not a jerk). Instead, here are the reasons that resonate with me:
1. Signed reviews are easier for authors to interpret. When I received my first signed review as an author of a manuscript, I realized how much easier it was to understand the reviewer's recommendations. By signing my reviews, I'm offering authors additional context into the comments I've provided. I include my name, title, department, university affiliation, and email, which makes me easily google-able (they could even find this blog post!).
2. Sometimes it's pretty obvious who the reviewer is anyway. On occasion, I've received reviews as an author that are almost laughably transparent as to the identity of the "anonymous" reviewer. The science world isn't so big! If you're a taxonomic expert in a particular group of organisms, it's easy to guess that it's you catching spelling mistakes in names of subfamilies (or suggesting someone cite your papers).
3. It's gratifying to be explicitly included in the process of improving a manuscript. I work hard to be a thorough reviewer, and I hope that my effort may be rewarded by a reputation for fairness and helpful insight. I appreciate my name being included in the acknowledgements (although this did once cause some consternation with a few peers, as it wasn't clear I was a reviewer and they thought I was collaborating with their competitors). In fact, I particularly enjoy reviewing for PeerJ, as my review is published with the early version of the manuscript.
Here are a few things to keep in mind about signing your reviews:
1. Be consistent. It's easy to sign a glowing review of a manuscript that you're accepting without recommendations. It's harder to sign a review of a manuscript you're rejecting, but (for the reasons I mention above), is arguably more important for both you and the author.
2. Ask journals before agreeing to review. Here's a tweet exchange I had with @rmflight about corresponding with journals (pure genius!):
3. You get used to it quickly. Once I committed to my decision to sign all reviews (for papers I accept AND reject), it only took a few reviews to feel comfortable with it. Nothing disastrous has happened, and reviewing papers feels less like working for free.@k8hert that is why I ask the editor first before agreeing to review, and make it a condition of reviewing— Robert M Flight (@rmflight) January 15, 2016
4. You can do it! Anonymous peer review is still the default in most areas of academia. In most cases, though, signing is a choice. I've heard early career scientists warned away from the practice, but it hopefully helps you to know that there is, in fact, a community of folks who make transparency in science a priority, and they exist in all career stages.
In short, I've signed ALL my reviews for about five years now. In that time, I've published my own papers, applied for grants, and even landed a tenure track faculty job I'm still quite happy with my decision.
(Thanks to @MusselDS for asking the question and prompting me to finally write this post.)