It's not complete bullshit - lots of toxic chemicals are lipid-soluable and get stored into adipose fatty-tissue along with the fat.  The liver is an amazing organ, but there are some things (heavy metals, some organic-associated compounds, etc) that the liver cannot break down and the kidney cannot selectively pass through to the bladder for evacuation.
It's definitely not the case that (so far as I know) there's been any solid work to show that these fat-soluable toxins do a lot of damage, though, so the execution of this drastic fat-storage-emptying-by-fasting will produce any kind of real result is... iffy.
But it's not completely unfounded in principle.  
This does beg the question of why not just lose the weight a more healthy, happy way than sudden and violent withholding of nutrients.
A steady decline of calorie consumption (calorie restricted diet, not a total fast) has DEFINITELY been shown to prolong life, both in mammals and non-mammalian vertebrates, and almost certainly would have the same effects in a human body... but this is due to a decrease in metabolic flux and insulin based cell-signaling, resulting in less DNA damage from reactive oxygen species formed during catabolic break down of energy-rich molecules as well as less cell stress from insulin response, not anything to do with "detoxing your fat."