How to make a function to play only on the home page?

I feel like I’m missing a lot of context from your question. If you are only asking how to execute a script on the homepage (and not on other pages), you can use the is_front_page() function. You can wrap your entire script in an if statement, like below:

<?php if ( is_front_page() ) : ?>

YOUR SCRIPT HERE

<?php endif; ?>

Check out https://developer.wordpress.org/reference/functions/is_front_page/ for more detailed information about this function.

EDIT

To only show the message on the front page, but keep the rest of the code on all other pages, you can simply wrap the message part in your conditional. Check out below:

<div id="cursorz">

  <?php if ( is_front_page() ) : ?>

    <div class="message" id="mess">Look at you! You look beautiful today!</div>

  <?php endif; ?>

</div>