(@ARGV               # if @ARGV has elements
  &&                 # and
    $ARGV[0] eq '-v' # The first element of @ARGV is '-v'
      &&             # and
        shift);      # remove the first element from @ARGV
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In Perl, && is a shortcutting operator. That is, in A && B, perl will only evaluate B if A is true. The first && in your code is a traditional logical AND but the second makes sure that shift is only executed if the first && is true (ie when @ARGV isn't empty and it's first element is '-v').

The logical result is then asssigned to $verbose: true IFF $ARGV[0] eq "-v" and has the side-effect of shifting @ARGV if it was.