Q: How could someone use my facetime account without my passwords?

You used it. That is what Apple is telling you. Unfortunately, you didn't provide the rest of the message.

Identifying fraudulent "phishing" email

Phishing & Other Suspicious Emails

Beware of Browser Pop-Up Tech Support, Phishing Scams

It's possible someone guessed your password or it could have been revealed somewhere (how do you manage passwords?). Depending on your security questions they may have been guessed?

Login to the Apple ID on iCloud.com & check for all authorised devices. Remove any unknown ones. Also consider enabling 2 step authentication to secure the Apple account

Frequently asked questions about two-step verification for Apple ID - Apple Support

Google & other services have similar 2 factor authentication - it can help prevent unauthorised access.

It could also be spam too, meaning the account was not accessed but the spammer simply hoped you would use the links in the message to investigate it. Do the links look legitimate? Don't follow links in any email that you didn't expect to receive.