Classes in PHP

What is the point in them?
People say it’s so you can use them on more than one page; why not just put the functions into a file and include the file?
I’d rather not substitute:

include("fetching.php");
function();

For:

include("fetching.php");
$fetching = whatever-it-is;
$fetching->function();

Or even:

function Blahde() {
global $fetching;
$fetching->function();
[...]

Argh.
Can anyone enlighten me?

Edit: And why have the code tags got that extra space under them?