The labels are on the messages if you apply them, but the messages will remain in the inbox too, and you won't see anything that hints that you applied any sort of label when viewing from the inbox.
If you delete them from the inbox, they will also be deleted from the labeled area too.
It's not like gmail labels, where you can label them and remove them from the inbox and just file them.
Yes, that is a bit annoying.
Just select the messages from the list at the top of the page by checking the boxes and then clicking the dropdown to apply the labels to all the ones you want in one shot.
If you try to do it from an individual message, it will just refer you to the top list where it will be automatically checked for you and you will be expected to apply the label from there.
Be really careful you don't click that delete button, or the checked messages will end up gone forever. There is no trash section in the personal message area that you can recover deleted messages from.