Hi Bchiu,
Basically, if you get low enough in frequency rooms ring like a big bell at modal frequencies. That is, instead of a very even, flat response like you might have outdoors or in an echo-less chamber, certain very narrow frequencies will have severe peaks, the equivalent of 20x the power output at those peaks than at other frequencies. You also get nulls, severe dips that cut the effective power by 20x or more, and these peaks and valleys are very dependent on seating position.
Bass traps de-energize those modal nodes. They reduce the effects of the peaks and the valleys and enable a DSP/room correction/EQ setup to finish the job. I have done demonstrations in my home with 2-way speakers, and you can totally hear the mud in the bass leave and the speakers appear to add an octave to the bass response when the bass traps come back into the room. The effect is just as good with the subwoofer. The best way to add lots of bass is to control the room first. Then you'll be amazed how good and deep a 6-7" woofer will go, and how much better and louder you can play your sub.
Bass in addition to moderating the peaks and nulls bass traps also reduce the effects of seating positions and speaker placement.
The best order is to get basss traps first and then use room correction/DSP/EQ. In the absence of anything else room correction is still better than nothing however.
MiniDSP si not a closed loop at all.

You are thinking very advanced stuff like what Meyers does in concert halls. They actually can adjust EQ curves from the music as people start filling up the halls. Very very cool stuff, and total overkill in a home.
With a home system you measure multiple locations, get an average, and then create compensating EQ curves. What IS real time about miniDSP is that you can upload your changes dynamically from your PC.
Best,
Erik