I recently came across a book called How to think like a mathematician, which covers some of the general principles behind reading, writing, and understanding undergraduate level mathematics.
I would consider this book to be a walkthrough of mathematical grammar. Are there any others that take a similar approach?
Perhaps "The Language of Mathematics: A Linguistic and Philosophical Investigation" by Mohan Ganesalingam? It focuses on how mathematicians actually use language in their writing from the perspective of a linguist.