Is it an acceptable practice to take off in a microburst?
The airfield in the video is KAFF, the Air Force Academy air base. Everybody you see in the video are air force cadets or retired Air Force pilots (in the tows). 

There is a credible thread about this over on Reddit where one commenter says:


  I'm very curious as to how you got this video, i was under the impression it never left the airfield.
  
  This happened in April 2014, this past year. The weather is absolute crap here, especially for soaring. Well... we get good weather sometimes. Anyway, it's not uncommon for those TG-16A's to go up with a 25 knot gust... But i digress.
  The cadets were pushing in because winds were out of limits and the weather was getting worse... and BAM! Microburst.
  
  This microburst hit right next to the airfield. The tower spotted it early, gave a verbal warning "look out..." and cadets are trained to do the following: grab a wing (glider) and turn broadside into the wind and put the spoilers out. The tows were not so easy... nor lucky. Their takeoff speed is about 50 knots, and none of them were powered up when they lifted off the ground, to give you an idea of how bad the wind was. Their only maneuver is to face into the wind and pray they dont actually take off. The tows that took off left for COS airport... it took another 30 minutes of holding gliders before the tower let the cadets start moving the gliders.
  
  As for taking off... 55 kt gusts are the highest the Academy has had in a long time. Considering there were a half-dozen other aircraft within 100 ft of the tows, along with people (i.e. cadets around/in the gliders) if he was moving too far from his position in the queue, the safest action is to get some altitude and try and leave the microburst. Or at the very least put some altitude and distance in between himself and the cadets and aircraft. You saw how slow the tows in the air were moving relative to the ground... those were HARSH winds.
  At about 0:20, you can see a cadet hanging onto the wing of a glider on the bottom of a screen. This wind was scary. I don't know if anyone was up at the time, but full tempo ops can be up to 5 tows and 8 gliders... on a standard afternoon training day 3 tows and 4 or 5 gliders is normal. It looks like they were already pushing the gliders to the hangar...


I believe the commenter is wrong in that the planes were not powered up, but everything else is accurate (relatively so, those Cub (?) tow planes can take off much less than 50kts).

From your question:


  Taking off against the wind seems impossible because where does it get its thrust? And with the wind also seems impossible because all the wind is traveling at the same speed as the airplane, which to me should make the airplane drop like a stone.


Taking off against (into) the wind is the most preferable way to get an airplane into the air. It reduces the amount of ground speed that you need in order to take off. Aircraft work according to the airflow over the wings, if the airflow is great enough and the angle of attack is right, then the aircraft flies. 

Taking off with the wind is also possible, but it extends your ground roll since you need to get to the same relative speed with respect to the wind before the airplane will fly.

Look at it this way, lets say you have the wind at your back and you take off on a bike. As you gain speed there is a point at which there will feel like there is no wind since you have the same speed as the wind at your back. As you keep going faster you'll start to feel like the wind is blowing at you, but slower. The amount of wind blowing at you is your air speed, the speed you are actually going is your ground speed.

Airplanes work the same way, taking off down wind requires the aircraft to get a greater ground speed (but the same air speed) to take off. After the airplane starts moving the only thing that really matters is the speed of the relative air flowing over the wings. When somebody says they need 50 kts to take off, they mean 50kts of airspeed, or 50kts of air flowing over the wings.



Now, a little commentary. There are two aircraft that take off in the video, one goes to the left which is up wind and the other goes to the right which is down wind. The upwind pilot has a much easier time controlling the aircraft since they don't need as much ground speed to keep it in control.

The downwind pilot almost bought the farm, and I'm pretty sure when he got back on the ground he needed a change of pants and a new aircraft seat. He's having a lot more difficulty controlling the aircraft because of the low airspeed. It doesn't look like he had much of a choice to go in that direction though, so I'd buy that man a beer if I ever met him just to hear the story.

The thing to take away from this is those pilots are retired Air Force pilots with thousands of hours of experience in high performance aircraft. The average pilot would be a stain on the tarmac in similar conditions. 



To answer your edited question:


  Is that standard practice to take off and fly to try to escape a burst? Is that even allowed and/or sensible?


No, it is not standard practice. Larger aircraft would probably not have this issue but the aircraft in the video all weigh less than 1000lbs, a little more with the pilots and fuel probably but not much. Pilots are trained specifically to avoid microbursts if possible because they are very dangerous things. A search of the NTSB accident database for the keyword "microburst" provides 73 accident reports with that mentioned, 21 with fatalities.

Is it allowed? No, airports will halt operations when low level wind shear is detected, however its not to say an aircraft isn't already in its takeoff run when the shear is detected. The highest priority is to stay on the ground. The aircraft in the video however don't have a choice, the wind speed they encountered is greater than their take-off speed, they were probably going to fly or tumbleweed.

CFR 91.3(b) gives a pilot full authority to deviate from any ATC instruction or rule in an emergency. An emergency does not need to be declared first, a pilots responsibility is Aviate, Navigate, Communicate. If all your attention is on flying (or keeping from flying) you can be in an emergency without telling ATC. I'm sure these pilots did not face any disciplinary or regulatory actions as the result of taking off from the parking ramp.