I've never used it before, but it looks fun! I think you should start small, then build up your grammar. I got rid of the square brackets around you month names. They are for regex character classes, if I understand this module correctly:

my $grammar = q{
    transaction:    date
    date:          /(Jan|Feb|Mar|Apr|May|Jun|Jul|Aug|Sep|Oct|Nov|Dec)\
+s+\d{1,2},/
};

# My output...
# $VAR1 = 'Oct 31,';
[download]

UPDATE: and a little more readable:

my $grammar = q{
    transaction:    date
    date:          /(
                        (Jan|Feb|Mar|Apr|May|Jun|Jul|Aug|Sep|Oct|Nov|D
+ec)   # Month
                        \s+ \d{1,2} , \s+ \d{4}                       
+      # Day, Year
                    )
                    \s+ (\d{1,2} : \d{1,2}) : \d{1,2} \s+ (A|P)M      
+      # Time
                    \s+ PCKLog \s+ log
                   /x
};
[download]