Hello enderk, and welcome to the Monastery!

What you’re overlooking is the line:

@words = @newwords;
[download]

which updates the contents of @words at the end of each loop iteration. Here’s one way of visualising what is happening:

use warnings;
use strict;

my @words = kmers(3);

sub kmers
{
    my $k     = shift;
    my @bases = qw( A T G C );
    my @words = @bases;

    for my $i (1 .. $k - 1)
    {
        print "\nBEGIN \$i = $i:\n";
        print "  \@bases = @bases\n";
        print "  \@words = @words\n";

        my @newwords;

        for my $w (@words)
        {
            push @newwords, $w.$_ for @bases;
        }

        @words = @newwords;

        print "END   \$i = $i:\n";
        print "  \@bases = @bases\n";
        print "  \@words = @words\n";
    }

    return @words;
}
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Output:

13:20 >perl 769_SoPW.pl

BEGIN $i = 1:
  @bases = A T G C
  @words = A T G C
END   $i = 1:
  @bases = A T G C
  @words = AA AT AG AC TA TT TG TC GA GT GG GC CA CT CG CC

BEGIN $i = 2:
  @bases = A T G C
  @words = AA AT AG AC TA TT TG TC GA GT GG GC CA CT CG CC
END   $i = 2:
  @bases = A T G C
  @words = AAA AAT AAG AAC ATA ATT ATG ATC AGA AGT AGG AGC ACA ACT ACG
+ ACC TAA TAT TAG TAC TTA TTT TTG TTC TGA TGT TGG TGC TCA TCT TCG TCC 
+GAA GAT GAG GAC GTA GTT GTG GTC GGA GGT GGG GGC GCA GCT GCG GCC CAA C
+AT CAG CAC CTA CTT CTG CTC CGA CGT CGG CGC CCA CCT CCG CCC

13:20 >
[download]

As you can see, on the second iteration ($i == 2), each element of @words is initially two characters in length; so, naturally, when it is concatenated with a base character, the result is three characters long, as required.

Hope that helps,