There are two ways to delete multiple emails at once.
First, you can click on the uppermost checkbox located below the Keep as New button. This will select all emails in your mailbox.
Second, you can select a range by clicking on the checkbox next to the first (or last) message, hold Shift, and then select the checkbox next to the last (or first) message. After selecting the desired range, simply click the Delete button.
However, you should delete them in lots of a few hundred at a time. If you try to delete all 32K+ at once it will most likely cause the mail application to freeze.