Rat-arsed means extremely drunk — so drunk you can barely stand or think straight. It is British, Irish, and Australian slang, and it is rude. You would not write it in a school essay or say it to a teacher. It grew out of older jokes like pissed as a rat and drunk as a rat: in British slang pissed already means drunk, and the rat is only there to make it stronger, the way drunk as a skunk does. There is no special story about rats drinking. Younger British speakers still use it; Americans mostly do not, and they may hear pissed as angry, not drunk. If you need politer English, say very drunk or wasted.