What does "\!" aka backslash exclamation mark do?

I don't know what the commands \! are supposed to do in the code you show, other than raising many errors.

The command \! is only allowed in math mode (unless redefined, which I'd discourage). It's purpose is to insert a negative thin space, which is useful in several places. For instance

\biggl(\frac{121}{12}-1\biggr)^{\!2}

is an improvement over ^{2}, because it moves the exponent towards the parenthesis and takes care of its bending.

Another place where it is helpful is in 2/\!\log x, because for technical reasons / is an ordinary symbol and TeX would add a thin space between it and the “log” operator.

Here's a picture: left the output with \!, right without.

The negative spacing of \! exactly matches the positive one by \, (which is automatically inserted in some places) in math mode. Note that \, can be used outside of math mode, where it inserts a sixth of a quad of space.

If you want to tighten a table, reduce the size of \tabcolsep. Using explicit negative spaces all over the place is not the correct way.

And never use \resizebox on a table.

This symbol is usually defined as

 \! negative thin space (normally 1/6 of a quad)

It is commonly used in tables, and formulas among other ways to manipulate spaces in LaTeX

And before you do post a question, please do run a search with similar keywords on this site. Hope this helps.