What exactly does Perl's "bless" do?

In general, bless associates an object with a class.

package MyClass;
my $object = { };
bless $object, "MyClass";

Now when you invoke a method on $object, Perl know which package to search for the method.

If the second argument is omitted, as in your example, the current package/class is used.

For the sake of clarity, your example might be written as follows:

sub new { 
  my $class = shift; 
  my $self = { }; 
  bless $self, $class; 
} 

EDIT: See kixx's good answer for a little more detail.

bless associates a reference with a package.

It doesn't matter what the reference is to, it can be to a hash (most common case), to an array (not so common), to a scalar (usually this indicates an inside-out object), to a regular expression, subroutine or TYPEGLOB (see the book Object Oriented Perl: A Comprehensive Guide to Concepts and Programming Techniques by Damian Conway for useful examples) or even a reference to a file or directory handle (least common case).

The effect bless-ing has is that it allows you to apply special syntax to the blessed reference.

For example, if a blessed reference is stored in $obj (associated by bless with package "Class"), then $obj->foo(@args) will call a subroutine foo and pass as first argument the reference $obj followed by the rest of the arguments (@args). The subroutine should be defined in package "Class". If there is no subroutine foo in package "Class", a list of other packages (taken form the array @ISA in the package "Class") will be searched and the first subroutine foo found will be called.

Short version: it's marking that hash as attached to the current package namespace (so that that package provides its class implementation).