How can I attach a hard drive to an Apple IIe?

First, you need a controller and drive. SCSI controllers are getting kind of collectible, so expect to pay upwards of US$100 for one on eBay. With some diligence, you might get lucky and find one for free in someone's attic-bound Apple.

A contemporary vendor sells a controller card that connects a CompactFlash card to an Apple called the CFFA:
http://dreher.net/CFforAppleII/
While this costs as much as one of the collectible SCSI cards, it has the advantage of being physically new and free of the problems of old mechanical hard drives just waiting for their turn to fail.

And how can I load applications on it? Most Apple
IIe programs are DOS 3.3 and you cannot format a hard
drive in DOS 3.3.

That can be kind of tricky. There are lots of options (communications programs, transfer programs, etc). It all kind of depends on what you want to copy over. If you want to just launch games, DOS launchers like this would help:
http://www.apple2.org.za/gswv/a2zine/System/DOS33LaunchInfo.txt

Do I format the hard drive in ProDos?

Yes, you will be formatting it with ProDOS. Your SCSI card will have associated software for partitioning the physical drive. If it doesn't come with the card you find, it is generally find-able on the internet. The CFFA card imposes fixed partition sizes on the CompactFlash card, so there's nothing to be done there.

How do I set up a hard drive in ProDos format? Do I
just create a menu manually?

Once the drive is partitioned, you format it with the normal ProDOS menu system.