This addition to the myriad terms for being drunk entered the language in Britain in the 1990s. It is sometimes shortened just to 'ratted'. It is a follow-on to the earlier phrase 'as pissed as a rat'. There wasn't any particular reason to pick on rats, the choice seems almost arbitrary. Other creatures which have been used in similar phrases are newts, ticks, skunks - even lords:
As drunk as a lord
As drunk as a skunk
As pissed as a newt
As tight as a tick
The first example I can find of the term in print is from a piece in The Guardian (Manchester), October 1992:
"The white Rhodesians acquired the undying admiration of their political allies in South Africa, so they thought, for their courageous defiance of world opinion in the face of sanctions. Actually, what really impressed them was that they defied the world even though most of them appeared to be rat-arsed from shortly after breakfast."