This book has a long history and was initiated by a seminar given by the second author and a paper he wrote. The response was so positive that it was made into a book that has known several translations and is used as a textbook for courses worldwide. Since the sixth edition Barbara Gastel has joined in and with this new edition, she has become the first author since Robert Day has been emeritus for a while now. The evolution of digital publishing has revolutionized the scientific publishing landscape, which made a new revised edition necessary (the previous one is from 2011). New items are for example the ORCID (that is a unique digital identifier distinguishing an author from any other researcher), the archiving of your (published) paper, warnings against predatory journals, digital poster presentations. There is also a new chapter on editing your own work before publishing, which is somehow a summary of what has already been said in previous chapters.
Barbara Gastel has a position in biomedical sciences, and Robert Day was teaching English. Their backgrounds clearly show in the focus they have put in the book. Even though there are many general guidelines on how to disseminate your research results, and it certainly introduces inexperienced students to the whole process of publishing a research paper, in my opinion the book is very much oriented towards the habits viable in life sciences and there is an emphasis on writing correct English. I am not familiar with publication culture in social sciences, but as far as mathematics is concerned, one should, after reading this book, consult additionally some books or papers that focus on the peculiarities of publishing mathematics. You may find tens if not hundreds of relevant papers and guidelines available on the Web. Most probably your own institution has one. And then there are of course some of the "classics" listed at the bottom of this review.
Another issue with this book is that this is mainly written for native English speakers/writers and to some extent even for English speaking American students. This is relevant to understand the humor and word play that is included (there are some old and new cartoons for which language is less relevant). Although there is a section on "writing in English as a foreign language", the emphasis there and in the rest of the book is to use correct English sentences and grammar. In fact a lot of attention goes to grammatical issues and common errors. This of course is important in writing in general, hence also in scientific writing
Another repeated pattern that is used throughout the book is the IMRAD (Introduction, Methods, Results, and Discussion) structure in all writings or presentation, whether it is a paper, a thesis, a seminar, a report, or whatever. To some extent this also applies to mathematical papers, but not as strictly as it is in life sciences. This format is much more suited for empirical papers, and in some journals publishing experimental results, especially in chemistry or biomedical journals, these five words are actually used as section titles. The results section is the main thing and many of the technicalities or formulas are sometimes banned from the paper and are added as "additional material" provided elsewhere. This is not exactly how a mathematical paper should be composed.
So in this book, guidelines on how to write your paper are following the IMRAD paradigm. After discussing the title, the list of authors, their addresses and emails, the I, M (here referring to Methods and Materials), R, and D sections are discussed separately, and the paper is finished with appropriate acknowledgements and references. The intended readership is obviously the community of students who did bot publish before, so the whole process is explained including the selection of a journal, submitting your paper, the refereeing, and how to react to it, and finally the postrefereeing stage of proofreading and publishing. Clearly, besides all the recommendations given, most journals have specific guidelines for authors that should be consulted. This is repeatedly stressed in the book as well. But the book is covering a very broad publishing culture, by discussing also review papers, and letters to the editors, or writing for a general public, composing a conference abstract or report, or how to prepare a poster or an oral presentation, or write a thesis or a project proposal. Once you became an established author you probably are already familiar with how to write a peer review, but there is still some advice given here. Also how to write a book review, give an interview, or write a book proposal. And for the really ambitious, how to become a science communicator.
So there are many general guidelines on writing. Certainly the part on writing correct English is extensive but not exactly connected to science writing. There are no particular guidelines for writing mathematical papers. The only place where mathematics is explicitly mentioned is when it is discussed in what order the authors should be listed. It is said that sometimes the order is alphabetical "like for example in mathematics". It is almost standard that mathematical papers are written in LaTeX and somewhat less generally accepted that references are managed with BibTeX. These tools are not even mentioned. The role of arXiv, Zentralblatt, and MathSciNet and the Mathematical Subject Classification (MSC) are not discussed. Neither do they mention the UDC classification. But the book is not only about writing or communication in a strict sense, there is also a discussion about ethics, plagiarism but the pitfalls of selfplagiarism are not highlighted. In this respect, I cannot resist to mention this tendency to multiply the number of papers using a process that is somewhat stimulated by this IMRAD structure: just modify the method or the materials and repeat what has been done already in other papers, and in this way you can produce many carbon copies of just one skeleton paper. Such an objectionable publication policy is less common in mathematics, but it can be a problem for numerical computing papers where a slight variation in the method or the equation to which it is applied can duplicate existing papers. This is of course the consequence of the equally disputable policy of evaluating researchers by counting their papers. In this perspective, it is also remarkable that the book does not discuss impact factors. There is only a distinction between "primary" and "secondary" publications. The impact factors for biomedical journals are so much larger than for mathematical journals that this may be a lesser issue there. Anyway, impact factors reduce mathematics to a negligible section in the science publishing landscape. The authors restrict themselves to give as many generally applicable practical guidelines as possible, but they rightfully avoid points that may raise some controversy since such discussion need not be included in an (under)graduate course. Another recent issue that is not discussed is the data life cycle management (DLM) which should ensure that data, results, software, etc. are still available in the long run. An issue of quickly rising importance in a digital age of fake news.
The book ends with several appendices. The first appendix is a list of abbreviations for words used in journal titles. "Math." is there, but "Comput." for "computer" or "computational" is not. For the benefit of mathematical students I should mention here the useful AMS list of journals and abbreviations. The second appendix is a list of jargon words to be avoided with a preferred alternative. That is certainly useful, also for mathematics. Next are lists of magnitudes (from atto to exa) and of many helpful websites, a glossary of terms used in the book, and an extensive list of references (but the ones below are not in the list), and finally a subject index.
Conclusion: there are a lot of general guidelines for undergraduates who never published a paper before to learn about the process. Especially the guidelines for using correct English are quite useful. For mathematics one may want to read some extra, more specific, guidelines. Of course, as is also mentioned in this book, much can be learned by consulting (good) examples and by imitation.
Some references relevant for mathematical writing

N. Steenrod, P. Halmos, M. Schiffer, J. Dieudonné How to Write Mathematics l'Enseignement Mathématique, vol. 16, 1970 and AMS booklet 1973
in particular P. Halmos' paper of about 30 pages is still recommended.  S. Krantz, A Primer on Mathematical Writing. AMS, 1996. A second updated edition was published by the AMS in 2017.
 S. Krantz. Mathematical Publishing, A Guidebook. AMS, 2005.
 S. Krantz. How to write your first paper. Notices of the AMS, vol. 54, no. 11, 2007.
 N. Higham. Handbook of Writing for the Mathematical Sciences. SIAM, 1998.
 T. Tao. On Writing. blog, retrieved October 10, 2017.