Click on the box to the left of the first email you want to delete and assuming you want to delete all the emails that are below that you can either press on your keyboard "shift" key and "PgDn" key repeatedly and it will mark all the other emails with a tick or use the "shift" key and the down arrow. This will also mark all the boxes with a tick or tick the first box you want to delete and holding the shift key move to the last email you want to delete and tick its box and release the shift key. All the emails in between should now be ticked.
If you don't all the emails that have been marked with a tick deleted you need to go down the list and untick the ones you want to keep.
Once you have done that you then click on the "Delete" button in the menu bar which should delete them on mass however if it does not work, once they have all been ticked, you should try to drag and drop them into the trash folder.
You may need do this a number of times due to the volume of emails that you want to delete at one time.
You could temporarily download a mail client such as Thunderbird, configure it as POP3 ensuring that 'Leave a copy of the message on the server' is not ticked. This will download all the mails and delete them from the server. You can then simply just delete the client. Move any you wish to keep to a different folder before downloading.