my @b = grep EXPR, @a;
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is equivalent to

my @b;
foreach (@a) {
   if (EXPR) {
      push @b, $_;
   }
}
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In this case:

my @uniqueIPs;
foreach (@uniqueIPs) {
   if (!$counts{$_}++) {
      push @ipAddresses, $_;
   }
}
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As for post-incrementing,

$x++
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is equivalent to

my $orig_x = $x;
++$x;
$orig_x
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In this case:

my @uniqueIPs;
foreach (@uniqueIPs) {
   my $old_count = $counts{$_};
   $counts{$_}++;             # Add one to count.
   if (!$old_count) {         # If it's the first time we've seen it,
      push @ipAddresses, $_;  #    save it
   }
}
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So,

my %counts;
my @uniqueIPs = grep !$counts{$_}++, @ipAddresses;
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is short for

my @uniqueIPs;
foreach (@uniqueIPs) {
   $counts{$_}++;             # Add one to count.
   if ($counts{$_} == 1) {    # If it's the first time we've seen it,
      push @ipAddresses, $_;  #    save it
   }
}
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It is dense, but it's tried and true. Just add a comment for the less learned readers.

# Remove duplicates IP addresses by counting
# the number of times each address occurs.
my %counts;
my @uniqueIPs = grep !$counts{$_}++, @ipAddresses;
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Oh by the way, you could also write it as follows if it's less confusing:

# Remove duplicates IP addresses by counting
# the number of times each address occurs.
my %counts;
my @uniqueIPs = grep ++$counts{$_} == 1, @ipAddresses;
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