The invoice number is used to verify the sale for publishers. Many publishers of paid asset or plugin ask for the invoice number upon request of support. The invoice number is also used as a license key to activate some asset or plugin.
The invoice number can be found in two place:
After you bought the asset, you will be sent an email whose subject is "Unity Asset Store purchase confirmation...". The invoice number is in the PDF attachment of this email.
Open https://www.assetstore.unity3d.com/#!/account/transactions, then you can find the invoice number in the Description column.