This package registers a /webmention route that accepts incoming webmentions. After validation, source and target are stored in the webmentions table, to be processed asynchronously. (How that happens is up to you; typically, you'd fetch the source URL, parse it for microformats, and somehow map those to an instance of, e.g., a Comment model.)
It also lets you send webmentions from anywhere in your Laravel app, like so:
use janboddez\Webmention\WebmentionSender;
...
// Somewhere in your app, like after a post is first published.
$result = WebmentionSender::send($source, $target);
$result is an associated array containing the target, endpoint, result (true or false), HTTP status code, and timestamp.
(WebmentionSender::send() will look for a Webmention endpoint on your behalf, no worries.)
Or, even better, have it automatically discover any links in your post content:
$results = [];
foreach (WebmentionSender::findLinks($post->content) as $target) {
$results[] = WebmentionSender::send($post->permalink, $target);
}