Headcanon is a truly disgusting, putrid concept. Actually that's not quite fair.
Why Headcanon is Disgusting
It's often based on nostalgia and fads rather than good writing. The most tragic example of this would be the shitty writing of the show that happened over the 1980s, which were written by people who hated everything done during the Tom Baker years, and hated Graham Williams and Douglas Adams for being funny (because God forbid a show starring comedy actors ever be funny), and hated Philip Hinchcliffe and Robert Holmes for ruining their headcanons about the Time Lords. They expected the Time Lords to be perfect monks with Godly Powers. Perhaps some of them are exactly this in more ways than one, but The Deadly Assassin depicted another side to them entirely. It hardly ruined Doctor Who as a show whatsoever (granted it didn't improve it either; contrary to what fans believe, depiction is never as important as plot), but the fans sure thought so. So a lot of the work produced over the later half of that decade basically existed to undo the "mistake" of making the Doctor not the only important being the Universe and making the Time Lords mortals, so they worked hard to make the Doctor God himself, and the Time Lords gods as well.
What they thought they were doing was being noble heroes fighting the money-machine and restoring art (they thought that Philip Hinchcliffe's decisions were only to make those evil scrubs with *le gasp* new opinions like the show to reap money from them, because shows should never be written to acquire new fans. Oh no. Everybody knows that shows really exist only to enable dying fanbases to slowly rot in absolute stagnation).
...What these were really doing was etching away the cosmicism that once defined the show bit by bit as they made the Doctor and the Time Lords more and more important and all of those truly Weird Tales beings the show introduced less and less important, and thus, ironically, erasing the actual artistic elements of the show. Furthermore, they put in excessive amounts of adult material because these were often fans of the show during the Lloyd era who really wanted to show to be written exclusively for adults, so massive amounts of violent and adult content that would make Hinchcliffe himself blush was placed on the show.
Their efforts to fight the windmills they mistook for giants got so bad that Michael Grade had to cancel the show. Thus, the fanboys brought about something far worse than the Doctor not being important enough.
This is proof that fanboys - and burnouts from the sixties, for that matter - should never, ever, be allowed to write for books, comics, audios, and telly. Believe it or not, shows don't exist for one particular generation of fans. They exist to do new things to acquire new fans, meaning that your fucking shitty headcanons will be risked, and if you can't handle that...
Their headcanons sucked, and so do yours!
.....It will be a dark, dark day when Tennant fangirls get to write the show. A dark day.
Headcanon was originally used to sort out retarded, incomprehensible canons. This is pretty much needed if you want to make any sense of the Doctor Who Universe. The only other methods are living in denial or detaching yourselves from the universe and going back to being a member of civilised society.
If you accepted the third option you may still encounter Headcanon for artistic reasons such as ambigous endings, implied info too subtle to get through TARDIS Data core's thick skulls or viewer interpretations in general. This interpretation is toxic and impossible to grasp for Tumblr people. Leading to posts like this (see right):
The best headcanons are ones which actively erase shit from Doctor Who. This can include devaluing a Doctor you don't like by saying the previous Doctor's regeneration went wrong to explaining away stories to entire eras as dreams. This version is pretty petty but so is everything else on the site.
BASED WHITAKER/5 Headcanons
Headcanon has since been abused and corrupted by the worst kind of Tumblrites to basically just make up their own shit and give it some kind of flimsy pretense of validity. The most egregious example comes from this blog: http://lgbtdisneyheadcanons.tumblr.com/ This blog isn't even related to Doctor Who (yet) but it serves as the perfect demonstration as to how you can basically make your fanfiction canon by just using your head, regardless of actual information showed, character consistency or unfortunate implications. This gets particularly irritating in Doctor Who considering how stupid the Tumblr section of Who fandom is.
Literally Doctor Who on the Planet of Autism.