@BIBS Webmaster ...
'simplest terms ...' ... it's broke ... but you know that! ;)
Suggestion ... probably best not to piggy back on someone else's posting ... start your own thread. That way your problem/issue gets the attention it deserves. Diff ... you have remotely hosted - Linux. Shared or dedicated server?
Did you set up cron job and is it running as it should? Yep ... some email out is related to cron job.
Did you set up an SMTP server for relay of mail? (some providers ... maybe Fasthost ... allow no port 25 out and have issues with PHPMailer [built info Moodle code] so do you see anything in server logs (OS logs ... not Moodles) that might give a hint/clue?
As admin user do you get Notification EMails about updates to Moodle? If, so, can you look at the full header of that message to see how it was delivered.
Anything in your SPAM folder?
https://docs.moodle.org/37/en/Mail_configuration
https://docs.moodle.org/37/en/Debugging#Debug_email_sending
Sigh! Wish is was 'simple' and 'non-technical' ... but ... it ain't!
'SoS', Ken