There are people around here more qualified to discuss some of the guts, but I can give you a rough idea of what the article is suggesting. What they are saying is that Perl's algorithm handles repeated backtracking poorly.

#!/usr/bin/perl
use strict;
use warnings;
use Time::HiRes qw(gettimeofday);

for my $n (1 .. 25) {
    my @start = gettimeofday;
    
    $_ = 'a' x $n;
    /(?:a?){$n}a{$n}/;
    
    my @end = gettimeofday;
    
    my $elapsed = $end[0] - $start[0] + 10**-6 * ($end[1] - $start[1])
+;
    
    print "$elapsed\n";
}
[download]

outputs

4.5e-005
3.2e-005
3.1e-005
3.4e-005
3.6e-005
4.2e-005
5.4e-005
8.1e-005
0.00013
0.000232
0.000448
0.000899
0.001696
0.003383
0.006309
0.012884
0.026508
0.053743
0.108439
0.212322
0.431355
0.890586
1.703086
3.468719
7.17187
[download]

Frankly, while I'm sure there is room for improvement in the regex library, I would characterize their example as poorly written code - rather than using repeated one-off matches, you should be using variable-width matches:

#!/usr/bin/perl
use strict;
use warnings;
use Time::HiRes qw(gettimeofday);

for my $n (1 .. 25) {
    my @start = gettimeofday;
    
    $_ = 'a' x $n;
    /a{0,$n}a{$n}/;
    
    my @end = gettimeofday;
    
    my $elapsed = $end[0] - $start[0] + 10**-6 * ($end[1] - $start[1])
+;
    
    print "$elapsed\n";
}
[download]

outputs

4.2e-005
2.9e-005
2.8e-005
2.9e-005
2.8e-005
2.9e-005
2.9e-005
2.9e-005
2.9e-005
2.8e-005
2.8e-005
2.9e-005
2.9e-005
2.9e-005
2.9e-005
2.9e-005
2.9e-005
2.9e-005
2.9e-005
3e-005
2.9e-005
2.9e-005
3e-005
3e-005
2.9e-005
[download]

and does the same thing. Note I've used curly brackets ({}) to indicate repetition; see perlretut for more info.