Speaking of insults: "sod off!" meaning and origin

I've thought it had to do with the love that dare not speak its name, to put it ever so coyly, but what does this phrase mean and connote?

And what's the approximate American equivalent?

Here's Eric Partridge from the Dict. of Slang and Unconv. English:

sod. A sodomist: low coll.: Mid-C. 19-20; ob.-2. Hence, a pejorative, orig. and gen. violent: late C. 19-20. Often used in ignorance of its origin: cf. bugger.

So your sense of "sod" is on the money. Suffixial "off" marks a general epithet as an insult, as seen in "piss off," "f-ck off," "bugger off," etc., all used in the manner of "please go away." Also note that "sod" in this sense is UK usage only. In US English it only refers to topsoil and turf grass.