This expression first appeared in the 1788 second edition of A Classical Dictionary of the Vulgar Tongue, written by British antiquarian Francis Grose. At that time, the phrase "piss-proud" originally referred to a penis made erect ("proud" meaning both "swollen" and "full of pride") by a full bladder. During sleep, the accumulation of urine in the bladder places pressure on the prostate gland, which exacerbates the natural tendency for males to wake up with an erection. Grose defined the term as meaning one who has a false erection, usually "said of any old fellow who marries a young wife".
Over the years, the meaning of the phrase shifted from the literal to the figurative and now, instead of denoting a man with an erection caused by bladder pressure rather than virility, it instead describes someone with an ego inflated past the point of any ability he or she has to back it up, or someone who is full of himself. Incidentally, the phrase "full of himself" appears to be directly related to "piss-proud" in that the former is a euphemism for the latter.
The most common remedy for someone being "piss-proud" is for a person, usually one who is socially close to him or her, to "take the piss".
Sources
BBC - h2g2 - The Origins and Common Usage of British Swear-words
<http://www.bbc.co.uk/dna/h2g2/A753527>
World Wide Worlds <http://www.quinion.com/words/qa/qa-tak2.htm>
knot.magazine <http://www.knotmag.com/?article=984>