How do I check that my back-in-stock emails actually went out?

Open Plugins → In-Stock Notifier and check:

1. Subscribers tab: search the customer's Email. Waiting means they haven't been emailed for that product yet; Notified means the email went out, and the Notified column shows the date.
2. Settings tab: Show the notify-me form on out-of-stock products and Back-in-stock email must both be ticked, or nothing is sent on restock.

If they're still Waiting after the restock, click Notify on their row. If they show Notified, ask them to check their spam folder. You can also look in Settings → Web Settings → Email Delivery Logs, which lists the emails your store sent.