Why does a European supplier (exporter) need an export declaration?

If a European exporter sells goods to a customer outside the EU, and the final delivery address is outside of the EU, the exporter has the right to sell without VAT.
The exporter has to prove to local tax authorities that they have the right to sell without VAT and must add a copy of the export declaration or exit confirmation document issued by European customs to their bookkeeping (Some European customs authorities do not issue a separate exit confirmation document). There are also many other reasons why customs authorities want exporters to do an export declaration (export control, statistics etc.).
The transport company has to ensure that it has the export document to cross the EU border.