The reason why Zayd is seen as the rightful Imam is because of his belief that a true Imam must fight corrupt regimes, which is exactly what he seeked to acheive in his rebellion against the Umayyads. Unlike Zayd, his brother Muhammad al-Baqi was more quiet and did not engage in any political action.
Unlike other Shia groups, the Zaidis do not believe in the infallibility of the Imams nor that they are divine. Zaidis also do not believe that the Imamate must pass from father to son, but believe it can be held by any descendent of Ali. They also reject the Twelver notion of a hidden Imam, and like the Ismailis believe in a living imam, or even imams. Doctrine wise, the Zaidis are similar to the Sunni Mutazilite.
Zaidis form the dominant religious group in Yemen, and the leader of the Zaidi community took the title of Caliph. As such, the ruler of Yemen was known as the Caliph, and this system continued until the middle of the 20th century.