Let me start by saying that the Mixtec area is definitely one of the least studied areas in Mesoamerica, yet Mixtec writing (primarily the codices) is the one thing that people have actually studied in great detail, but there is still a great deal of questions. As far as we can tell, it seems like Mixtec writing developed during the Classic period as a derivative of Zapotec writing but as a local phenomenon that we called the Ñuiñe style. However, there are also some elements that hint at possible central Mexico connections, primarily with Teotihuacan, yet at this moment they are purely speculative since not much is known about Teotihuacan influence on the Mixtec area. Ñuiñe writing is primarily comprised of calendrical dates that probably represent the names of important figures like rulers or nobles. Some writing denotes the names of places where important rituals are taking place like offerings and human sacrifice. However, this may say more about the corpus of evidence where we have found Ñuiñe writing than about how it was actually used.
By the Postclassic period, Mixtec writing as we see in the codices seems to have developed along similar lines as other central Mexican scripts in that it was mainly logographic, meaning that images represent complete words or ideas rather than specific sounds. However, as we will see with Aztec writing later, Mixtecs also used phonetic devices to emphasized specific readings, primarily during the spelling of place glyphs.
That being say, something that I want to emphasized is that, contrary to popular beliefs, you do not need complex fully functioning scripts that allow you to write everything about a language in order to have complex socio-political systems. For the Mixtecs, writing was more about making political statements for a selected audience than about bureaucratic record keeping. In fact, this is true for a lot of societies in Mesoamerica including the Maya that even with a logo-syllabic script, it was never used for bureaucratic purposes as we see in other places like the old world. Ñuiñe writing and the Mixtec in the codices functioned perfectly for the purposes that they were meant.