I will base this answer on what I remember from school - we discussed Büchner extensively, but then this was more than 30 years ago. So this will be a stand-in until somebody more knowledgable gives a better answer.
Politically, Büchner is important because he lived, and actively took part in, auspicious times. This is now known as the "Vormärz" (pre-march), the time before the abortive March revolution of 1848. Despite the reforms forced by the Napoleonic occupation Germany was not a unified state. Politically interested Germans, who had come to learn about the ideals of the French revolution, started to dream of a united German national state. The main factions where the Liberals, who imagined a constitutional monarchy with a strong middle class, and the Democrats, who placed a lot more emphasis on equality for all the state's subjects. Either way, this was a direct challenge to the power of regional potentates, and the would-be revolutionaries suffered political persecution.
Büchner belonged to the Democratic faction. His "Hessischer Landbote" was not nilly-willy social criticism, even in the heavily edited form it was eventually published in it was quite literally a declaration of war upon feudal power (after all it proclaimed "war upon the palaces"). The idea of a united German state with its citizens as sovereign was progressive to a point where basically every established German power tried to stop it, and Büchner was a preeiminent voice of that idea (which, incidentally, is now enshrined as a founding principle in the German constitution).
As far as form is concerned, he was important because he dispensed with the romanticism of his time in favor of social realism, replaced stylized with colloquial language and featured lower-class characters (supporting in "Danton's Death", main character in "Woyzeck") who where not comic relief, but psychologically fully fleshed out (Büchner was, after all, a trained scientist, although in anatomy and medicine rather than psychology; still, I guess that is how he honed his observational skills).
Having said that, there is a certain romantic aura that surrounds the person of Büchner himself. He was, after all, a brave, youthful rebel who eloquently took on the ruling powers, and had nothing but an early death to show for it - a sort of Kurt Cobain of his time, even if that was obviously not a sentiment from his contemporaries. He has also the rare advantage (for a writer who is taught in schools) that he is quite appealing to young people - "I feel crushed under the horrible fatalism of history" is a sentiment every teenager can relate to who ever thought that they could build a better world by next Tuesday, if those pesky old people would not get in the way all the time.
His work is also accessible because it is short (you can read all of the collected Büchner in a couple of days) and internally consistent - Büchner did not live long enough to revise any of his opinions, and there are not two diametrically opposed phases (no "Sturm und Drang" vs. "classic Büchner") that are hardly reconcilable.