What is the purpose of the transaction failed email?

Allows you to recover failed credit card charges.

Not all rejects are fraudulent, though most declines can be readily rejected for various known characteristics, some are due to cvv or billing address issues and are valid.

Also, if your gateway goes down, you can recover the orders that could have been placed.

Call the customer and vet the order out for validity.

Basically a backup for the store owner.

I believe these are sent to the Store owner. This is taken from the Magento website:

The Payment Failed Emails section allows you to configure the email template to be used, the recipient, as well as the sender of the email notification messages that are sent whenever a customer transactions fail.

To configure Payment Failed Email:

In the Payment Failed Email Receiver list, select the email address that notifications will be sent to when a payment fails.

In the Payment Failed Email Sender list, select the email address that notifications will be sent from when a payment fails.

In the Payment Failed Templates list, select the email template to be used for email notification of failed transactions.

In the Send Payment Failed Email Copy To field, type any other email addresses that you want to receive a copy of the notification. To include multiple addresses, separate each by a comma.

In the Send Payment Failed Copy Method field, select one of the following:

Bcc: (Blind Carbon Copy) Sends a copy of the email message, but does not include the email address of the recipient in the email header.

Separate Email: Sends a separate copy of the email notification to the recipient, and includes the email address of the recipient in the message header.

Source: http://www.magentocommerce.com/knowledge-base/entry/configuration-sales-checkout