The first step is to make it so that you can prove that you own your site without sharing anything like your Neocities login.
The easiest way to do this is by using an email you can check.
<head> section of your page, add a link to this email like:
<link href="mailto:email@example.com" rel="me">
The second step is to keep track of what webmentions you've been sent. There are lots of different ways to do this but the simplest is webmention.io. You can run it yourself if you want to but it's free to use it here.
Since you've added that email link, it can look at your site, see that email, then send it a link to login.
After you follow the signup process you'll have a username for your site on webmention.io. It's okay if it's not the same as the neocities name. You then add another link in that
<head> section, replacing "username" with the username you just got:
<link rel="webmention" href="https://webmention.io/username/webmention" />
This will mean that when someone wants to send you a webmention, they'll send it to webmention.io, and webmention.io will keep track of it for you.
showing your webmentions: webmention.js
There are lots of ways to make it so that you can send out webmentions automatically when you publish new stuff, but I recommend sending them manually at first to get a feel for it.
First, add a link on one of your pages to somebody who supports webmentions. If you're just testing, try adding a link to this site; you can use it to test receiving webmentions too.
Then go to webmention.app and put the url of your page, the one with the link, into the box at the bottom. It will look at your page, see all the links, go to those links, and see which ones support webmentions. It will then let you send webmentions to those sites.
You can then go back to the testing site to see if they got your mention!