...but I don't really quite get the idea of callbacks, and particularly I don't understand the syntax.

Well, it's not just quite like with a phone - you ring up somebody, they hang up and call you - but rather a way to parametrize a subroutines execution chain via a subroutine reference (like a function pointer). The called sub may have been compiled into any context, and the compilation rules of that context apply. And what ikegami said.

I don't really get the first line of the get_callback routine either, when nothing appears to be being passed across

It gets something passed across but discards it.

A sub routine called as a method ($thingy->routine(@args)) by something ($thingy - irrelevant whether $thingy is a literal, a blessed reference, a scalar holding a string, or an expression which resolves to a package or a blessed reference) gets passed the invocant as its first parameter in @_.

So, having this preamble

sub routine { 
   my($thingy, $tx) = @_; # $thingy is not used
   print "routine: $tx\n";
}

my $ref = \&routine; # take a reference to the sub
my $pkg = "main";
my $obj = bless do { \my $x };

# we could also say...
#   my $pkg = __PACKAGE__;
# ...to get any package we are in in compile time
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the following call expressions are all equivalent:

# method calls

main->routine("foo");
main->$ref   ("foo");
$pkg->routine("foo");
$pkg->$ref   ("foo");
$obj->routine("foo");
$obj->$ref   ("foo");
# or even
(join'',map{chr}(109,97,105,110))->routine("foo");
__PACKAGE__->$ref("foo");
(bless \my $x)->$ref("foo");

# function calls

routine ('main', "foo");
$ref -> ('main', "foo");
routine ($pkg,   "foo");
$ref -> ($pkg,   "foo");
&$ref   ($pkg,   "foo");

routine ($obj,   "foo");
# etc.
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More on differences between subroutine calls either as function or method: perlsub, perlmod, perlref.

The expression

my (undef, $thing) = @_;
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is just a fancy way of saying

shift @_; # drop first element of argument list
my ($thing) = @_;
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The subroutine get_callback you quoted just seems not to care at all about it's invocant, but nonetheless expects to be called as a method. Hmm, code smell...