The main reason is the sheer power and huge amounts of damage it deals to the opposing Pokemon. 130 base power, even though it's 10 off of last Gen's 140, is still extremely powerful and will leave a dent in pretty much anything assuming that it isn't a fairy. Latias and Latios are both different Pokemon to each other and use it for different reasons (specifically in X and Y), so I think I'll answer each Pokemon separately.
Latias
Latias is more defensive than Latios is. It still has 110 Special Attack though, but since Latios has 130 in that stat, it generally outclasses Latias offensively. So why does Latias use Draco Meteor? Because it still deals high damage, and the Special Attack drop matters much less to it.
Since Latias is often more interested in its support moves than its attacking moves, it will most likely use Draco Meteor once and then switch out if it uses Draco Meteor at all (Psyshock is the favourable option since it hits more Pokemon super effectively). So if it limits its use of the move in this way, it won't really have to worry about the drawback as much.
Still though, Latias doesn't always use Draco Meteor. Sometimes it may be replaced by a useful move like Defog or Calm Mind.
According to sweepercalc, 60% of Latias use Draco Meteor on Pokemon Showdown.
Latios
Latios will almost always use Draco Meteor. Why? Because there is another thing it almost always uses: Choice Specs and Choice Scarf.
Latios does not really have the greatest setup moves with Calm Mind pretty much being the best of the bunch, so generally it will prefer to use Specs just so it can have immediate power. Any defensive or support roll is outclassed by Latias or another Pokemon, so that means Latios will most likely be using a choice item.
This use of choice items makes Draco Meteor a practical choice. Since choice Pokemon get locked into their move, quite often the opponent will switch into something that resist Dragon or Psychic attacks (Steel type and bulky Fairies being common). This means that even if it didn't get the Special Attack drop, it will have to switch out, meaning the drop didn't matter at all.
If the opponent does not switch, whatever the opponent is using will take a huge chunk of damage, which will either KO it or leave it vulnerable to you next Pokemon's move. Heck, even with the Special Attack drop, another Draco Meteor would probably knock it out!
According to sweepercalc, 83% of Latios use Draco Meteor on Pokemon Showdown.
