Connect Moneybird under Company settings → Integrations. Nothing is pushed to Moneybird: the connection is only used to recognise your existing Moneybird contacts.
When a customer is created in ReserveBee, ReserveBee looks that person up in your connected Moneybird administration by email address. If a contact with the same email exists, the customer is automatically added to the customer group you picked under Default customer group (if you haven't picked one, a group called "Moneybird" is used).
Pair that group with a group-restricted Variant and your Moneybird contacts get their own pricing without you assigning anyone by hand.
The lookup runs once, when the customer is first created. It isn't an ongoing sync.
Moneybird is a Professional-plan feature.