It's not a backdoor in the existing router firmware. It's an article about how to add a backdoor to the firmware and then re-flash the router with that firmware, which requires authenticated access.

Oh, oops, I upvoted this because I thought there was a backdoor in the Linksys WRT54g (original title: "Linksys wrt54g backdoor"). Instead it's just about developing custom firmware. I shall look before voting, next time...

I don't understand how this qualifies as a backdoor if you need to flash the router to gain access to it.

Isn't that putting the chicken before the egg?