Actually, grep !/^\./ should not be considered as stated. grep in Perl is not similar to the grep family of utilities in the Unix word. While often, it uses regular expression as part of the block or expression, that's the only similitude.
The proper way of thinking of grep is as this:

Code:

grep BLOCK LIST
grep EXPRESSION,LIST

Notice the coma.

The first will be seen as:

Code:

my @list_result = grep {!/^\./} @given_list;

Inside that block {} you can put a lot of normal code, in this case is a regular expression and the return of that expression is negated. If that evaluates to true, the content of $_ (which contains an element of the list @given_list) is appended to @list_result.

The second will be seen as:

Code:

my @list_result = grep !/^\./, @given_list;

or

Code:

my @list_result = grep (!/^\./, @given_list);

The difference between BLOCK and EXPRESSION is that the later only accepts one expression instead of whole block of code. For this purpose, it is the same.

Going back to this portion:

Code:

@fails = grep /FileNotFoundException\|I\/O exception\|fail$/, ( grep !/^\./, readdir LOGDIR );

First, you need to eliminate those escape characters highlighted in red, otherwise you are just making the | a normal pipe character and not the regex alternation, indicating OR
I\/O that needs to be escaped only because you are using the default delimiters m// and when the code gets parsed it sees the first / it reads until I/ and it thinks is done, and it does not know what to do with the rest.
If the default delimiter is used, the m is optional, but it is not optional if another character is used as delimiter.
Can you now see what that line of code is? Can you tell if it is a grep BLOCK LIST or a grep EXPRESSION,LIST?
It is a nested set of grep EXPRESSION,LIST.
The first one is grep !/^\./, readdir LOGDIR which will return a LIST of the filenames in LOGDIR that do not start with a period, being itself the LIST argument passed to the second grep /FileNotFoundException|I\/O exception|fail$/, (...) represented by the three dots.

As a note, the last grep is not reading the content of the files and searching for the strings FileNotFoundException or I/O exception or fail$. It's using those regex against the filename, if any of those filenames match in the name, that element gets appended to @fails.