Very violently. The Safavid conversion took place literally over the span of one person's rule. That person was Shah Ismail I. Before him, Persia was predominantly Sunni and considered one of the centers of Hanafi learning (although the Safavids themselves were initially Shafi'i Sunnis). In order to give his country a distinct identity from the Ottomans (who were Sunni), Shah Ismail I basically forced his kingdom into Shi'ism almost overnight. He destroyed Sunni mosques, began state-sponsored ritual cursing of the first three caliphs (whom 12 Shi'ites consider to be usurpers), disbanded Sufi tariqas, and expelled any Sunni scholar who wouldn't convert. He also began a holiday celebrating the assassination of the second caliph Umar.

Later rulers such as Shah Ismail II were less anti-Sunni and it wasn't really until Muhammad Baqir Majlisi, a Shi'ite cleric, that Shi'ism became almost synonymous with Persia.

The Safavids were as much of a product of the people they ruled, as any major polity of that time. With your question, I'm going to assume you mean the height of the Safavids coinciding with the rule of Suleiman the Magnificent and his father Selim in the Ottoman Empire. Shi'a ideas were not strange in the common theological debates of the day. The Fatamids before them, in Greater Syria and Egypt, were also Shi'a, and sent many missionaries out across the eastern Mediterranean. But now to the point of your question, "How did the Safavids MAKE people into Shi'is?" First, it needs to be realized that divisions like "Sunni" and "Shi'a" didn't exist in that world, these are qualifiers that were applied later. While it's true, the Safavids promulgated a "Shi'i" ideology and the Ottomans a "Sunni" one, the dividers between the two lay along political matters as well as religious ones. The Safavids MADE people Shi'i by denouncing those who were not as traitors. If you betray the state religion, you betray the state.

edit: Source can be found in Lapidus' chapter on the Safavids in his excellent work "Islamic Societies to the Nineteenth Century