XML::Twig uses select internally to redirect print output which is mucking up your use of print - that's getting redirected too. You could work around it by using STDOUT explicitly in yout diagnostic print:

use strict;
use warnings;
use XML::Twig;

my $outStr = '';
open my $outFile, '>', \$outStr;

my $twig = XML::Twig->new (
    twig_roots => {'wibble' => sub {put_new_jump (@_, $outFile, 'wibbl
+e')}},
    twig_print_outside_roots => $outFile,
    keep_encoding            => 1,
    );

$twig->parse (<<XML);
<root>
    <wibble name='My Name is Mr. Wibble'>The wibble element</wibble>
</root>
XML

close $outFile;

print "All done now. Result is:\n\n";
print $outStr;


sub put_new_jump {
    my ($tree, $elem, $fh, $path) = @_;

    my $name = $elem->att ('name');

    print STDOUT "Name:  $name\n";
    $elem->print ($fh);
}
[download]

Prints:

Name:  My Name is Mr. Wibble
All done now. Result is:

<root>
    <wibble name="My Name is Mr. Wibble">The wibble element</wibble>
</root>
[download]