The idea behind the extended primary is that it gives your yeast time to clean up any undesired fermentation by-products and drop clear. There is a certain logic to it and it seems fairly reasonable given the old belief was the 1-2-3 system of one week in primary, two in secondary and three in the bottle (another belief worth deconstructing itself). If you are spending three weeks between primary and secondary, an additional week in primary isn't that big of a deal. The idea gained steam when some homebrewers left some beers in primary for an extra week or two and decided their beers tasted better that way. Then the whole thing snowballed as the new belief challenging the old primary and secondary paradigm.
The big criticism usually came from the supporters of the secondary-is-a-must camp that if you left the beer on the yeast you will get autolysis and your beer will taste like beef broth. There's a lot of misinformation on autolysis out there but suffice to say I have lambic two years in on the trub and it definitely has no beef broth flavor. I've left big Belgians, with no brett to mask autolysis flavor, in primary for three months with zero beef flavor. I can appreciate that commercial brewers do worry about autolysis. The difference is that they are fermenting in enormous cylindro-conical fermenters with substantially more beer on top of the trub. The combination of pressure from all that beer plus the conical shape results in a greater concentration of pressure and heat that can accelerate yeast death. Lambic brewers ferment right in those barrels and that's a 59 gallon primary where the trub might sit for years. Still a bigger system than an ale pail or carboy. Autolysis isn't necessarily a myth but it's pretty hard to treat brewing yeast so poorly at the homebrewing level you're going to get noticeable flavor from autolysis after a few weeks.
There are a couple reasons why that extended primary is a good thing. First, if you are brewing a bigger beer you need more aging time no matter what vessel it is in. Even a beer in the higher single digit ABV can benefit from some extended aging, whether it is in primary, secondary or lagered. Many commercial brewers do put their bigger beers through some extended aging and/or lagering for clarity and smoothness. Even ales. So there's that. Second, if you are mistreating your beer the extended time will help clean up some off-flavors that never should have been there in the first place. If you are stressing your yeast by under-pitching, under-aerating, or supplying insufficient nutrients, you risk creating off-flavors in your beer. Yeast will continue to take up some of the byproducts over time and fix your problems. So if you are doing that, then yes, you need the extra time. However, the more effective solution is to treat your beer right in the beginning.
Some beers are harmed by the extended aging. This is particularly true of weizen beers because the esters and fresh wheat flavors fade over time and rather than having a good mix of wheat, clove and banana you end up with sort of a bland clovey beer. The same is true for hop flavor/aroma-forward beers. Those late addition hops fade over time so unnecessary aging can drain out some of the flavor and aroma. Some beers, especially lower ABV beers, do have a certain freshness early on that gets lost over time, even week by week.
That's not to say you can't enjoy your beer more as it ages. All us beer snobs cellar beer because the flavors develop over time. Some higher ABV beers, as I said before, really mellow with age. Hell, I just knocked off the end of two 10%+ ABV Belgians that were two and a half years old that didn't mellow for over a year. Jamil Zainasheff had a medal winning BDSA after aging four years. The distinction I draw here is that if you treat your fermentation process properly you don't need to age every beer four or six weeks and some beers will be worse off for it. Don't believe just because somebody copies and pastes pages of bullshit over and over on forums about how you need to wait eight months for your beer to be done makes it truth.
If you want to do an extended aging because you think it makes your beer taste better, that's ok. It is your beer and you should drink it how you like it. My belief is that you should understand why you are doing it and do it for a particular purpose, not because somebody told you that you should. It is also my belief you should make the best beer possible and do so by employing the best processes out there, not whatever will cover up other mistakes.