"What is the reason for the first argument that is passed to the routine in a class being the class name? Is this useful in certain ways?"

You need the class name so that you can call other methods from within your method. For example:

{
   package Processor;
   sub process_one_thing {
      my $class = shift;
      my ($thing) = @_;
      print "Processing '$thing'...\n";
   }
   sub process_many_things {
      my $class = shift;
      my (@things) = @_;
      for my $thing (@things) {
          # we need $class so we can call this
          # other method!
          $class->process_one_thing($thing);
      }
   }
}

Processor->process_many_things("Foo", "Bar", "Baz");
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"I noticed that if I export my functions, from the class, and then, in my calling script call it by its name (e.g &function("data") ) then the first argument isn't the class name"

That's because you're not supposed to do that. Don't export methods ^*.

Basically methods in Perl are just syntactic sugar. Foo->bar(...) is a shorthand for Foo::bar("Foo", ...), but with inheritance thrown in, so that if there is no such function Foo::bar(), the parent classes of Foo will be consulted.

* it is sometimes possible to use exported functions and OO programming in the same package to good effect, but until you're comfortable using OO, don't try it.