Final answer:
The Pope's anointing of Pepin and his sons reinforced the Carolingian dynasty's legitimacy and established an alliance that created the Papal States, defining a new era of church-state relations and geopolitics in medieval Europe.
Step-by-step explanation:
The anointing of Pepin and his sons by the Pope had a significant two-fold effect on the Frankish kingdom and the broader stage of medieval European politics. Firstly, it legitimized the Carolingian dynasty over the Merovingians, cementing their authority and enabling Pepin to establish his rule with the blessing of the church. Secondly, it had a geopolitical impact, as this alliance between the Papacy and the Frankish rulers led to the Donation of Pepin, which laid the foundation for the Papal States and allowed the Papacy to become a territorial power independent of the Byzantine Empire. This anointing also facilitated cooperation and potential conflicts between future popes and Carolingian emperors, as seen when Charlemagne, Pepin's son, was crowned Emperor of the Romans by Pope Leo III, creating friction with the Byzantine rulers.