Neither a computer scientist, nor a documentation writer. I'm a humanities student! But I think this is what he means.
When writing in ordinary LaTeX (ie outside the verbatim environment), we use the grave accent, also known as the backtick (`), to open apostrophe or speech mark pairs. Meanwhile, we use the apostrophe (’) or the typewriter apostrophe (') to close. This produces the correct curly quotes when put through LaTeX. It therefore does not cause a problem, as exemplified by the following:

Butterick is not a fan, however, of the the LaTeX practice of continuing to use the grave accent to replace opening apostrophes and speech marks when put into the verbatim or coding environment. Most writers would present the following:
\documentclass{article}
\begin{document}
\section*{Incorrect usage}
\begin{verbatim}
if(``James Bond'' = trustworthy)
smile
if(`turkey-bacon' = bacon)
cry
\end{verbatim}
\end{document}

Notice how the result produces curly apostrophes. Butterick argues that this is bad practice because the code is meant to have typewriter apostrophes. When writing in code, you want to have the original symbol because it serves a unique functional purpose as part of the code, so to use a different symbol is to be inaccurate. This is in contrast to ordinary LaTeX, where the grave accent and typewriter apostrophe are just the curly quotes' typographically poor stand-ins, specifically designed to be replaced during compilation by better-looking alternatives. Even if you try to use a typewriter apostrophe, LaTeX gives you a curly closing apostrophe instead.
You can easily solve these concerns by:
- Loading the
upquote package (documentation) with \usepackage{upquote}; and
- Replacing any grave accents in your code with apostrophes
Applying those two steps to our existing example:
\documentclass{article}
\usepackage{upquote}
\begin{document}
\section*{Incorrect usage}
\begin{verbatim}
if(``James Bond'' = trustworthy)
smile
if(`turkey-bacon' = bacon)
cry
\end{verbatim}
\end{document}

This has the twin benefits of making the code exactly the same as you would want to copy into anything you were working on and removing any semantic ambiguity.
\documentclass{article} \begin{document} \verb+`test'+ \end{document}and you'll see what he's talking about. Related: http://tex.stackexchange.com/questions/166790/how-can-i-get-straight-double-quotes-in-listings – Paul Gessler May 14 '14 at 22:28