Here’s what it’s about, I think: Taqiyya is a Shi’ite thing. Muslims are supposed to proclaim their faith and never hide it. But the Shi’ites were the underdogs for much of Islamic history, and sometimes and in some places had to profess their beliefs underground, as it were; taqiyya told them it was all right to do so, if proclaiming Shi’a doctrine publicly would get them in trouble. That is all. Nothing about “Lie to those Sunni dogs so we can undermine them in secret from within!”, which is something like what RWs seem to think it means.
The concept – and the intra-Islamic controversy over it – really should not be all that hard for Westerners to understand. Christianity has faced similar problems in its history – from the start, in fact. It was an illegal, underground religion and Christians could hardly admit it publicly. Some even made the required public sacrifice to Rome or Caesar, just to get the magistrate off their backs. And even after Christianity was legalized, Christians were divided by the Donatist Controversy (If a priest, under state pressure, committed the mortal sins of apostasy and idolatry, were his sacraments still valid? The question was decided in the affirmative – a priest in a state of mortal sin remains a priest and can administer valid communions, absolutions, marriages, etc. Which has a reassuring relevance to contemporary Catholics, on which I will not here comment further.)