As Asiz's contractor and Vessel, the Flame Haze "Weaver of Coffins" Tis was dying, he sought to manifest in this world beside her while maintaining his preservation of her body using his ability Pure Coffin. Firstly he secured the Power of Existence needed to manifest by devouring the entire human army surrounding Tis, who were the ones who killed her. In the summoning ceremony Asiz acted as both the summoner and the one to be summoned, and he offered his connection to the Crimson Realm in order to summon and transfer himself out of his Vessel, Tis, and manifest into this world.
The aim of the Resummoning was to manifest in this world while maintaining Pure Coffin. If Asiz simply manifested normally, his Vessel, Tis, would break. If he returned to the Crimson Realm and then crossed back into this world as quickly as he could, Pure Coffin would cease functioning and Tis' body would be extinguished. With Resummoning, this process of crossing back and forth was omitted and Asiz instantly manifested himself as a Crimson Lord right next to Tis.
According to Merihim, Resummoning was a miraculous feat which only succeeded because Asiz was an extraordinary Unrestricted Spell Master, and also a miracle which only occurred because of the many humans who happened to be around him at the time. It was something which less skillful people, like Alastor, would not be able to accomplish.