Epigraphy is the study of inscriptions. There are many types of epigraphs; overtly political inscriptions like Prashashtis are only one of them. Land grant inscriptions, donative, and commemorative inscriptions, temple inscriptions, etc. give a good Peek into the art and culture of contemporary times.
Epigraphs can be classified according to their surface, language, content, geography, etc. Some of them would be royal, like Ashoka’s rock edicts, and some of them would be private, marking the burial locations of brave men. Even those inscriptions which were commissioned by the royalty give a peek into the culture, e.g. Allahabad Prashashti of Samudragupta was composed in classical Sanskrit.
Ashokan inscription at Lumbini gave us hints on the possible birthplace of Buddha. Then non-royal inscriptions like donative inscriptions in caves or stupas or temples give hints of the popularity of religious sects. Artworks had names of the artist inscribed on them.
Inscriptions commemorating brave sailors, women who committed Sati, ascetics who passed away in the grand tradition of Sallekhana, etc. tell us about the culture. They tell us what people at the time thought worthy of remembering for generations; what they valued. Land grant inscriptions like the Damodar copper plate inscription give us hints about agrarian relations, ongoing Bramhanisation-Sanskritisation of tribal population, etc.
Thus, inscriptions serve as sources to look at the entire society at the time; the political events and genealogies are only one of the many kinds of information that can be derived from them.