The three documents rarely agree perfectly: a supplier might invoice for a slightly different quantity than what arrived, or bill a price that does not match the original order. Three-way matching catches these mismatches automatically by comparing all three documents and only sending the exceptions to accounts payable, instead of a person checking every invoice by hand.
AI document tools have made this more practical for smaller finance teams, since the purchase order, receipt and invoice often arrive in different formats and from different systems. The common misconception is that a mismatch always means fraud. Most mismatches are ordinary errors, such as a partial shipment or a price update that has not been entered into the ERP yet.