On the case of Stephen: his father was the first Grand Prince of the Pagan Hungarians to be baptized and adopt Christianity (along with his wife, Sarolt who was baptized in Constantinople). In contrast he continued to allow Pagan rituals in his hold, so it was Stephen who was the truly first devout Christian of the Árpád dynasty. Géza also tried to establish the Christian monarchic law of primogeniture. After his his death however (around 997 a.d.), Koppány, also a descendant of Árpád, laid claim to the throne based on old Pagan traditions (mainly the oldest member of the dynasty taking the widow of the old Grand Prince as wife and ascend to Princedom).

This resulted in a rebellion by Koppány to overthrow Stephen and it was quite easy to interpret this armed conflict as a battle for the souls of the Hungarians (paganism vs. christianity). Koppány laid siege on the city of Veszprém (county capital to this day), where Stephen's army supplemented by German knights (settlers accompanying Gisela of Bavaria, Stephen's wife) came to the city's rescue. Koppány was defeated, his body was quartered and hung on the walls of 4 major stronghold (Veszprém, Esztergom, Győr and Gyulafehérvár).

After this conflict Stephen ascended to the throne as the first true christian king of Hungary. Accounts on his coronation are muddled at best, the most "mainstream" account was named the Hartvik legend (after Archbishop Hartvik written between 1095–1116 a.d.), where Archbishop Astrik was sent by St. Stephen to Rome to require or ask for a crown from the "Pope", but it doesn't mention a name by specifics. No matter how Astrik hurried the envoy of the Polish prince, Mieszko the First was faster, and the crown was prepared for the future Polish king. The pope had seen a dream during the night, where an angel from the Lord told him there will be another envoy from another nation, asking for a crown for their own king, and it should be them the crown should be given to. The next day Astrik approached the Pope and he received the Corona Latina.

However the Holy Crown is a actually a double crown, made of two separate pieces. The Corona Latina by Pope Sylvester the Second was used for his coronation, but later the Corona Graeca was added as an "expansion" to the crown. That makes the Holy Crown of Hungary the only crown blessed by both Catholic and Orthodox churches.

TL,DR: the pope did legitimize the rule of kings, but in Stephen's case it was "no-brainer" decision, as a devout catholic ruler fighting pagans and spreading the word of the Lord was far better than pagan "huns" raiding through Europe again.