Simple, its the least stressful of the roles.
Tanks are essentially the leaders of the group, and gotta know the routes and such, a failed mechanic by them can easily wipe the party, as a result they often get flamed the most when things go wrong.
Healers on the other hand gotta not only heal mechanics but also heal screw ups from players failing mechanics, all while also dealing with the mechanics themselves which complicates healing by forcing you to move.
A lot of DPS players also have multiple tank and healer alts. But you will never meet them because most of us refuse to pug. So unless you are in the friend group or guild you will just have to suffer the shortage. Perhaps if people were a little nicer.
I understand this is likely said with irony. But there are plenty of DPS with this mindset and then they wonder why they are vetted so hard.
This is untrue this season.
Nothing one shots you. Anymore. At least not in 11s.
Blizz has done a temendous job of making healing feel good.
Ironic as a casual I prefer tanking, but only do so for friends and guild. Now that I am WAY overgeared for it, I am also pugging heroics just for crests and stones, better drop rate. Every role has a job, just happens DPS’ job is the easiest:
Guess which one they do VERY well 
They have more fun and interesting rotations, better fantasy, it is competitive, and a higher skill ceiling.
Because DPS is the lowest pressure role. If don’t do a good job as a tank or a healer, the group fails. If you don’t do a good job as a DPS, you can still be carried. Tanks also have the extra pressure of leading the group.
I would add to what others have said is that there are just more chances for a dps play styles to grab a player. Blizzard has this problem in all of their games with the holy trinity by adding far more dps than the other two roles.
Because being a dps is the only way to show that you’re better than the guy next to you. And we all know that if you don’t have a big number in this game than you’re nothing.
=/
You have chosen… wisely. Also, probably the most honest answer I have ever seen.
DPS gets the biggest numbers.
Meanwhile Healers are expected to clean up everyone else’s mess and tanks are expected to be the leader.
Because big numbers causes a dopamine when dealing damage.
That’s how it is in most games. Most people choose the damage dealer option, even outside of WoW.
Because if Im forced to maximize my dps in every role, I figure I might as well play a spec that does a lot of damage without having to also worry about keeping people alive.
To be fair, to a lot of them it’s the entire game.
All the missing tanks are farming old content.
Because there is simply too much to know about encounters as a tank or healer, and the assumption is that people playing those roles know them.
Even as DPS at the LFR level I will watch videos beforehand to get an idea of mechanics, but nothing compares to actually playing the game and seeing them in action. I can learn on the fly as DPS (and to a lesser degree as healer) using my basic knowledge from a video; the tank cannot.
A learning tank means people are going to wipe and I don’t want to be responsible for that.
Gee, why do people wanna play the least stressful role 
Part of the issue is class restrictions. Some classes simply can’t heal/tank and I’d be happy to do the said role, but I simply don’t want to play any class besides warlock.
It’d be nice if someday blizzard adopted the elder scrolls online design where literally every class can use a weapon specialization that provides their own unique abilities and lets any class adopt any role in the game whether it be a tank, ranged/melee dps or healer.
They could also give warlocks on live the tank ability that season of discovery has. I am pretty jealous of that, but I prefer to stick with TWW.
Tanks’ and healers’ jobs are made significantly harder when other players mess up, not so much for damage dealers.