I restocked a product. Do I need to email the waiting list myself?

No. Subscribers are emailed automatically when the product's stock goes from zero back up, as long as Show the notify-me form on out-of-stock products and Back-in-stock email are both ticked in Plugins → In-Stock Notifier → Settings.

You can also send it yourself. On the Subscribers tab, find a row with the status Waiting and click Notify. That emails the people waiting for that product.

Once emailed, subscribers move to Notified and the date shows in the Notified column.