[^a-zA-Z\d\s:]

• \d - numeric class
• \s - whitespace
• a-zA-Z - matches all the letters
• ^ - negates them all - so you get - non numeric chars, non spaces and non colons

The [..] form is known as a collection (:h /[]) and can contain a sequence of characters and/or one or more ranges of characters, separated with -. It will match any single character that is contained in the sequence/range(s). You can, of course, also match the inverse with ^.

In :h white-space you'll find a collection equivalent to \w:

\w  word character:         [0-9A-Za-z_]

So one approach is to invert/negate that equivalent:

[^0-9A-Za-z_]

Alternatively, collections can also contain the "character classes" found under :h E944. These are the same character classes that have been around forever in the BRE/ERE patterns used by legacy tools and POSIX environments.

Alphanumeric chars have the following character class equivalent:

[:alnum:]   isalnum ASCII letters and digits

You can invert/negate that like so:

[^[:alnum:]]

(You could also probably use negative lookarounds with \w but that's relatively difficult compared to this straightforward analog to [^\w].)

Update: I just realized I didn't address the full pattern you posted. Just look for a character class that's equivalent to \s and you'll find [:space:]. And " is just ". So [^\w\s"] becomes:

[^[:alnum:][:space:]"]

One other thing, this is likely to be the most performant of any possible solutions since others have lookarounds or alternations (|) and those are expensive.

I would use the following Regular Expression: '[^a-zA-Z0-9]'

Pattern nonAlphanumeric = Pattern.compile('[^a-zA-Z0-9]');
Matcher matcher = nonAlphanumeric.matcher('Sales - Actual Results (TY)');
system.debug(matcher.replaceAll('')); // output: SalesActualResultsTY

Demo