« $&&&!$s++ » is « $& && !($s++) ».

!$s++ is a neat construct. It's true the first time it is evaluated, and only the first time it is evaluated.

$ perl -E'say !$s++ ? 1 : 0 for 1..5;'
1
0
0
0
0
[download]

It's often used to remove duplicates.

$ perl -E'say grep !$seen{$_}++, split //, "abracadabra";'
abrcd
[download]

As for $&, it is set by the match operator. The only match operator in the code is /^-+$/, so $& will be false until the first time that matches, and it will always be true afterwords.

The code prints the following lines:

• The second line of the file, unless the first line of the file consists entirely of dashes. [ I suspect this is unintentional and never happens. ]
• The line that comes after the first line of the file consisting entirely of dashes.
• The second line following a line consisting entirely of dashes, unless it or the line before it consists entirely of dashes.

I'm surprise the person who wrote that doesn't know about -n. He also left in unneeded spaces and parens despite his golfing attempt.

svmon -Pt15 | perl -e 'while(<>){print if($.==2||$&&&!$s++);$.=0 if(/^
+-+$/)}'

   |
   v

svmon -Pt15 | perl -ne'print if$.==2||$&&&!$s++;$.=0if/^-+$/'

svmon -Pt15 | perl -nlE'say if$.==2||$&&&!$s++;$.=0if/^-+$/'
[download]

while (defined($_ = <ARGV>)) {
    print $_ if $. == 2 or $& and not $s++;
    $. = 0 if /^-+$/;
}
[download]

• $&

It means that the author thought that a superficial appearance of cleverness was more important than doing his job properly - that is, writing stuff that is easily understood.