They seized control of a student movement that had become demoralized in the wake of the assassinations of King and Kennedy, the escalation of the Vietnam War and the election of Richard Nixon. Under the charismatic leadership of one-time lawyer Bernadine Dohrn and others, these modern day Jacobins declared a state of war against the U.S.
They bombed dozens of buildings, but they also suffered horribly at their own hands, most notoriously when a bomb they were building in New York's Greenwich Village exploded, killing several members. (Look carefully at the archival footage: That's next-door neighbor Dustin Hoffman gawking at the destruction.) Eventually, the war ended and the radicals were still in hiding, working menial jobs and quietly withering away.
The film does omit discussion of notable Weather Underground alumnus Kathy Boudin. After a decade on the run, Boudin, the daughter of a famous lefty lawyer, went to prison over twenty years ago for participating in a botched armored-car robbery that resulted in the murders of several cops. She was recently in the news, first when her son Chesa (who was raised by Dohrn and Ayers) won a Rhodes scholarship and, just a few weeks ago, when she was finally paroled over the vociferous objections of the New York law enforcement community.
One can only conclude, after seeing The Weather Underground, that the filmmakers were doing the still-incarcerated Boudin a big favor by leaving her out of this disturbing but very informative film.