How is TransferWise able to process international transfers so quickly?

This is usually achieved by something called Pre-funding. It is a concept by which the Sending company, pre-funds foreign currency with the beneficiary country's banks.

For example, if money has to move from the UK to the Philippines. Transferwise will maintain a pre-funded account with say Bank D'Oro (BDO) in the Philippines.

Once TW gets a confirmation of the payment in the UK, TW instructs BDO to release the equivalent amount in PHP (Philippines Pesos) into Beneficiary's account.

In this case, Transferwise is using its own balance placed in beneficiary country banks to fast-track the transfer.

The pre-funded accounts are topped up after every 3-5 days depending on the volume of transfers for each specific corridor.