Hello! I'm doing my MA in Arabic and Semitic philology, but I studied medieval Islamic history and Shi'ite theology during my Bachelor's.

Great question, let's see what was going there. You will undoubtedly be aware of the difference between Sunni and Shi‘a Islam, but I'll put them here just to provide the argument with more clarity.

 
Whence the difference between Shi‘ites and Sunnis?

The origins of the Shi‘a-Sunni split seem to be political in nature: after the death of Muhammad, there immediately arose a discussion on the subject of his succession; but it is very unlikely that either party called themselves by these names. When talking about the earliest proto-Shi‘ite movements, most scholars of medieval Islam prefer the term "Alid", to distinguish the largely political movement from the later theological movement.

In any case, the ‘Alid movement represented a minority, as it appears that most Muslims supported Abu Bakr as Muhammad's successor. Because of this, the ‘Alids became very popular with disenfranchised Muslims, at the period these were mostly converts from Christianity, Judaism, Zoroastrianism and particularly a host of millenialist cults and sects of the time. Heinz Halm argues that it is only from a later period onwards, around the time of the martyrdom of Husayn, that we can talk of a "theological" Shi'ism, which seems to have incorporated many ideas from Iranian religion(s), particularly on the concept of the "Light of Muhammad" (ar. nūr Muḥammad) and its understanding of hereditary authority.

 
The Shi'ite relation to the Abbasid revolution
Coming back to the point, as you can imagine the martyrdom of Ḥusayn at the hands of the Umayyads in Kerbala wasn't exactly warmly received by the Shi'ites. The reasons for the collapse of the Umayyads are many and cannot be discussed here, but it's obvious that the Abbasid revolution was at its core a populist grassroots movement, supported very strongly by Muslims of non-Arab origin, particularly from the area of Khorasan (roughly modern-day Turkmenistan).

Kennedy notes that "the problems of the [Muslim] community would never be resolved until the lead was taken by a member of the Holy Family", which is pretty much in line with basic Shi'ite ideas of rulership. Although the head of the ‘Abbasid dynasty was of the same descent as the candidate supported by the ‘Alids (‘Abd Allah b. Mu‘āwiya), he was of a different branch and not preferred by the majority of the ‘Alid party.

In the end, this difference led to open simultaneous revolts in Basra and Medina. Although the Medinan revolt was quickly crushed, the revolt in Basra spread initially rather successfully to other cities in Iraq, before it was finally crushed in Bākhamrā, on the road between Basra and Kūfa. The leader of the revolt died several days later of his wounds.

 
tl;dr: the ‘Alid (later Shi‘ite) community felt betrayed by the ‘Abbasids because they didn't choose their candidate to be the new leader

  Sources:

Kennedy, H. The Prophet and the Age of the Caliphates, 2004. Consult in particular the section on the early Abbasid caliphate.

Crone, P. Slaves on Horses: The Evolution of the Islamic Polity, 1980: 63-66.

 
Recommended reading:

Heinz Halm, Shi'ism, 2004. I'm pretty sure Halm has something about al-Saffāh, but I won't be home until January and I don't have the book in pdf. Gah!