How to quickly troubleshoot errors caused by missing closing brackets?

If you are using e-TeX (which you probably are), you can make TeX record grouping levels and strangenesses in the log file by

\tracinggroups=1
\tracingnesting=2

This can probably help to find the cause of an error if it is due to a grouping problem.

In your own code, it's probably best to give a short log message at the start of a group level so you can immediately identify in the log what groups are started and closed where.

Furthermore, you can show the current group nesting at any time in your code with

\showgroups

May I suggest you to use the syntonly package. It must be declared in the document header,

\usepackage{syntonly}

then, right after it you write

\syntaxonly

what it does, it checks only the syntax without compiling the document. It does not provide further help in spotting the errors, but at least you can avoid to compile the document unsuccessfully. Once the syntax is correct, just comment it.

%\syntaxonly

It's not nice, but for me debugging these types of latex errors, if you have a large document and really no idea where the wrong bracket structure might be, works best with the old "binary search".

Just comment out half of the document, compile, if it works, proceed with the commented part, otherwise with the uncommented part.

(If you \input files, proceed accordingly with the nested code.)

Seriously, call it stupid, but most of the time it's quickest to do exactly this, even though there are - theoretically - situations where you produce new errors by the commenting. Usually it's easy to see how to comment so it doesn't happen.

Of course, nicely to be combined with gcedo's answer.