What does the C ??!??! operator do?

As already stated ??!??! is essentially two trigraphs (??! and ??! again) mushed together that get replaced-translated to ||, i.e the logical OR, by the preprocessor.

The following table containing every trigraph should help disambiguate alternate trigraph combinations:

Trigraph   Replaces
??(        [
??)        ]
??<        {
??>        }
??/        \
??'        ^
??=        #
??!        |
??-        ~

^{Source: C: A Reference Manual 5th Edition}

So a trigraph that looks like ??(??) will eventually map to [], ??(??)??(??) will get replaced by [][] and so on, you get the idea.

Since trigraphs are substituted during preprocessing you could use cpp to get a view of the output yourself, using a silly trigr.c program:

void main(){ const char *s = "??!??!"; }

and processing it with:

cpp -trigraphs trigr.c

You'll get a console output of

void main(){ const char *s = "||"; }

As you can notice, the option -trigraphs must be specified or else cpp will issue a warning; this indicates how trigraphs are a thing of the past and of no modern value other than confusing people who might bump into them.

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As for the rationale behind the introduction of trigraphs, it is better understood when looking at the history section of ISO/IEC 646:

ISO/IEC 646 and its predecessor ASCII (ANSI X3.4) largely endorsed existing practice regarding character encodings in the telecommunications industry.

As ASCII did not provide a number of characters needed for languages other than English, a number of national variants were made that substituted some less-used characters with needed ones.

^{(emphasis mine)}

So, in essence, some needed characters (those for which a trigraph exists) were replaced in certain national variants. This leads to the alternate representation using trigraphs comprised of characters that other variants still had around.